The "Dad's Army" reputation of the home guard is extremely common and somewhat unfair, even if parts of it are based in truth. Even at the time the reputation was around, they were originally named LDV (Local Defence Volunteers) which was changed in conversation to Look, Duck and Vanish. This combined with LDV being uninspiring led to the name being changed to Home Guard a few months after their creation. Most of the problems the Home Guard faced were a combination of a few factors, mainly the unplanned nature of their implementation, the volume of enlistment and lack of materials to equip them.
Whereas civilian defence militias were fairly common on the continent, Britain historically had a volunteer military and due to the power of the Royal Navy it was extremely rare the the British Mainland was threatened with invasion. There was a sort of home guard in the First World War called the Volunteer Training Corps but this was largely based around preparing men for enlistment in the military rather than forming a defence force for the Home Isles. So when France fell in June 1940 there was no official defence force beyond that of the British Army, at a time where there was possibility of German invasion. From what i've read of War Office Memos (message me if you want me to dig them out) the government was initially divided on their policy towards this, we know Churchill was broadly in favour but British planning at the start of the war had not taken the prospect of German invasion seriously so little background work was done. The decision to create the Home Guard was also partly taken in response to public pressure. Some citizens were already putting together private forces for this purpose which made some official recognition inevitable.
But this raised another problem of what they were supposed to actually do. Most Home Guard volunteers fully expected to fight if a German invasion occurred, as opposed to the military's idea of them playing a more non combat support role. This led to little to no training in the early days of the Home Guard which did little to help their reputation.
The rapid growth was also a huge problem. When Eden put out the call for people to register for then LDV he expected about half a million volunteers, he ended up with 3 times that after a few months. All of a sudden the government has got to find weapons, uniforms and resources to train 3 times as many as expected and it simply did not have the resources available to be able to do it. After the evacuation at Dunkirk the British army army was short of on weapons and Britain was still transferring from a civilian footing to a war economy so the factories capable of mass producing millions of small arms quickly were not running yet. Demand far outstripped supply and of course the British Army received priority shipments of new weapons, this left very little for the Home Guard to be armed with. The government issued rifles as fast as it could but until 1943 many of the Home Guard were armed with whatever weapons they could get their hands on.
So it's easy to see where their reputation comes from. In the summer of 1940 we have over a million men who with limited weaponry, no uniform, little training, an unclear role, who were planning to take on the Wehrmacht in case of invasion. Added on this was the fact the membership was made up of people who were unfit for military services, mostly because they were too old but also for health reasons and those who served in reserved occupations. Added on this we have multiple fiasco with procurement, most famously the issue of hundreds of thousands of pikes in 1941 based on a misunderstanding of instruction from Churchill to arm the Home Guard.
This all being said, things got a lot better as the war progressed, and by 1943 they were an armed, uniformed force with basic training. We will never know how well they would have performed in the case of a German invasion, but we do know they provided a real contribution to the war effort. They freed up soldiers of the British army from many more mundane roles such as patrolling the Home Isles, manning check points etc. This allowed more of the British Army to be used in active combat roles.
It's easy to mock the Home Guard but i don't want to be too harsh, ultimately they were people who gave up their spare time for unpaid work because wanted to contribute to the defence of their country even though they couldn't join the military, and they did succeed in making a contribution.