Admittedly, my only knowledge was from reading an abridged version of Romance of the Three Kingdoms over a year ago and playing the Total War game. But stillll....
Romance makes it seem like total chaos! All the feudal lords were effectively independent warlords, making alliances and declaring war against each other at will. At the same time they paid lip service to the Emperor and valued Imperial gifts such as the "Nine Bestowments" and titles like "General of Chariots and Cavalry". Did these gifts and titles really mean anything?
It's crazy, it's like everyone knows there's no central power anymore but collectively agreed to play along like it still mattered. Did they even pay taxes to the imperial government anymore? Did the government have any real authority over the provinces and commanderies?
I just don't understand why anybody gave recognition to the imperial government when they were actively waging war against the prime minister at the same time. If the Emperor was so powerless that his "handler" could effectively rule in his name, then did the Emperor really matter? Why value was there in keeping up this farce?
To be honest, I'm not 100% sure if you are referring to pre three kingdoms as in pre 220 or pre 190 when asking about the structure though generally, the question seems pre 220.
3 parter
When games cover the parts before the Coalition vs Dong Zhuo (the Turban revolt of 184 or in RTK games, other years as well), they have a problem. Yes, it is a great narrative as it tells of the Han's decline and it foreshadows the rise of key figures (Cao family, Sun family, Liu Bei, Dong Zhuo) as they all fought in the Turban revolt, there are also expectations as the novel and Dynasty Warriors use the Turbans as an opener. However, it is Han, the one whole government vs the rebel (or rebels with a bit of shifting) so to make it a better game, the Han becomes Han court and (non-existent) warlords.
The structure of the Latter Han was the Son of Heaven at the top... sometimes. If young (as kept happening for the Han) then the Empress Dowager (from one of seven families till Emperor Ling went for He, who promptly got a lot of rumours attacking her background) and her relatives, led usually by the General-in-Chief (gave charge of capital troops and senior political ranking, most Han figures ignored the first part), would run the court. Even when the Emperor was an adult until the Emperor got killed or seized power back.
There was a central bureaucracy, 3 Excellencies sometimes with a Grand Tutor (appointed for a New Emperor) and sometimes a General-in-Chief, nine ministers and other key posts like the Bearer of the Gilded Mace in charge of security. An Imperial Secteriat to write the decrees, Cenosrate to keep an eye on everyone else and to act as the Emperor's eyes and ears (in theory), eunuchs to look after the inner apartments including the harem and politically act as an important buffer against the bureaucrats or overly powerful relatives. There were clerks, counsellors, attendants and so on.
At a provincial level, people from outside the area would be appointed as Grand Administrator of Commandries (or if under a powerless Han King, an imperial relative, then a chancellor instead who ran the area and supervised/kept an eye on said King) and they would be supervised by Inspectors (13, one for which province) who would issue reports of what was going on and, if a major revolt, raise troops across commandry lines. In the last few years, after the Turban revolt, the rank of Governor (which had been around in Former Han but was now revived on Liu Yan's urging) was made for frontier provinces to give extra authority of effective command, rather than being a role of oversight, over the provinces
The outsiders directly appointed by the court were meant to act as a check against the powerful local interests though in practice, being from the same backgrounds and maybe needing their help in future (or their having armed retainers) meant the outsiders didn't always act as an effective counter to said powers. Local officials would serve on their staff and at their lower levels.
In terms of military might, if possibly the local officials dealt with it with their local troops and raising armies as required. If going to be too large, a General or someone from the Censorate could be sent with permission to raise troops from nearby provinces and backed up by the Northern Army and it's five regiments, the only professional fighting force in the land. If close enough to borders, troops hired from nearby tribes like the Wuhuan might also reinforce the army. With the only professional troops, the imperial armoury and the resources of the state behind them, the Han army was usually strong enough to deal with internal threats. Frontier Garrisons under Commandants with a mix of professionals, convicts and volunteers supported by camps near Chang'an and one at Liyang helped protect the northern frontier.
The Han Emperor's authority was never absolute. Several had to plot to take power from their relatives, the eunuchs became a powerful force after three times helping Emperor's overcome their General-in-chief, laws were not always obeyed, the Han struggled to gain full access to the wealth of the empire due to a broken system or to always get people to serve or to stop murdering each other.
Yet the Han Emperor raised the proverbial flag against his controllers, the controller's position would often collapse. If the Han Emperor ordered an execution then only flight into exile might stop it, the Han Emperor could force things through with enough effort. A dynasty centuries old, the concept of the Son of Heaven and that the Han held the mandate, was powerful. One might protest against the policies, about who the Emperor favoured, one might turn a blind eye as many did to regicide but the Emperor was still the Emperor, the Han was still the Han, the only legitimate rulers.
I would like to correct that these feudal lords you mentioned only came after the Yellow Turban Rebellion, by which the (Eastern) Han dynasty was already entering its final years.
Han dynasty's bureaucracy was largely similar to Qin dynasty, its precursor.
They have this system of 三公九卿, 3 excellencies and 9 ministers, who are the basis of the central government.
Specifically for Eastern Han, the dynasty before Three Kingdoms, the three excellencies are
The 9 ministers are
For local governments (outside of the capital and immediate regions), there are 3 broad levels in the following descending order: Provinces (州,zhou), Commanderies (郡,jun), and Counties ( (縣 ,xian) .
The 'head' of a province is called 刺史, Inspector. As the name implies, their only jobs is to supervise and inspect the works of officials in the commanderies and counties under him. If unsatisfactory work or foul play is found, they have the rights to impeach them through writing reports to the central government. However, they have no actual political or military control of the provinces they are in charged of, and their salary is only 600 dan.
外十二州，每州刺史一人，六百石。
Those 12 provinces outside the capital, each has an inspector, salary of 600 dan (yearly)
《后汉书·百官五》
* 1 dan is about 30kg worth of grains
Comparatively, the 'head' of a commandery, called 太守, Grand Administrator (GA) has a salary of 2000 dan.
每郡置太守一人，二千石
Every commandery has one GA of salary 2000 dan
GAs are the ones with the most local power within a commandery.
Next up, we have counties. Sometimes there are also 國 、邑、道s, but they are of the same level as counties. 國 are kingdoms of male royal members or refer to marquessates (houguo 侯國) . Han practice a grading system for statuses below kings with 20 levels, the twentieth being a full marquess with a marquessate. 邑 are fiefs of princesses. 道 are counties with majority non-Han ethnic people.
Counties have 縣 令 , County Prefects, as the highest authority.
Kingdoms/Marquessates have 相, Chancellors as the equivalent. Kings/Marquess have no direct authority over the land they 'own', they are only entitled to a portion of taxes collected.
Inspectors, GAs, County Prefects and Chancellors are all appointed by the central government. Kings were able to appoint all other officials in his kingdom.
If you want to know more about the whole governance structure, check out the book The Bureaucracy of Han Times (cambridge.org)
Now after this brief overview, we can go back to your question.
These 'warlords' you mentioned largely have control of 1 entire province (at least, originally). Like Liu Biao for Jing Zhou (荆州). In a sense, they are essentially inspectors. So what caused inspectors who originally have no actual power become so influential?
As time passes, some inspectors and GAs began to work together for profit. The inspectors will only write good things in their reports about the GAs of his province, and in return, GAs give benefits to the inspectors, or even work for these inspectors to help them establish control of their province.
This caused evident problems and unhappiness among commoners who are often disregarded or even mistreated. This became one of the drivers for the aforementioned Yellow Turban Rebellion.
When these rebellions came, especially in faraway places the Han army could not reach quickly, inspectors often get their fellow GAs to recruit local armies to defend against rebels, which the central government acquiesced. Especially since most of these inspectors did indeed managed to defeat the rebels.
Following the death of rebel leaders in 185 AD, there were still some remnant rebels that lead uprisings across China. These uprisings were not as significant as the original Yellow Turbans, but nonetheless an annoyance.
One court official hence suggested that
刺史、太守，货赂为官，割剥百姓，以致离叛。可选清名重臣以为牧伯，镇安方夏
Inspectors and GAs are corrupted and mistreat the people for their profits, leading to such conflicts and rebels. (we) can appoint key officials with good morals/records as
牧, Governor, of a province to have better control and resolve these issues, bringing peace to the empire.《三国志·刘焉传》
This new governor essentially has the combined powers of an inspector and a GA.
The emperor agreed and appointed a few inspectors to become the new 'governor'. For inspectors that didn't become governors officially, they were also allowed (implicitly) to keep their military for handling the occasional rebellions. Those who succeeded in suppressing the rebels can also become a governor.
From here on, the power of local officials only went up, leading to people like Cao Cao having their own armies and being able to go against the central government when Dong Zhuo rose to power.
In essence, what you observed was only a common occurrence in the last part of Han dynasty history.
Why keep up the pretense of a functional government when everyone was practically independent?
Well there could be lots of reasons.
The things you mentioned that were given, like the 9 bestowments, are symbols of authority and power. They do have symbolic meanings that of value.
Recognizing the central government was also a way for your power in the provinces to be legitimatized. In the sense that you are not just another annoying rebel but someone officially appointed by the holy emperor to manage this area.
Perhaps someone else could add more to this.