During the entire Interregnum, Cromwell struggled to manage Royalist sympathies even within his own camp. Many parliamentarians leading up to and during the civil wars never foresaw a political system that did not include a monarch of some kind. Most were simply opposed to any monarch with Catholic sympathies or the Stuart line in general, they were fully prepared to accept a hard-line Protestant monarch. However much like the French Revolution, events escalated too quickly for them and it became unpopular to remain moderate towards the monarchy. The New Model Army in particular became the core of republican sentiment within the parliamentary faction. Since this army was so successful at crushing Royalist and regional uprisings, the new political system was essentially built around them and their ideology, with Oliver Cromwell, one of their most popular officers, at the helm.

Cromwell made repeated efforts to enact republican/democratic forms of government but was almost always frustrated by the delays, incompetence and divisions in the parliaments he called. He would suddenly dissolve these parliaments and resort to centralising all political power around himself and a cadre of military officers. Cromwell eventually consolidated more power than any monarch previously had and it's possible to view his rule as a military dictatorship rather than a republican government. However, you have to consider that Cromwell was a fierce puritan with a temperate personality, he held his title of Lord Protector with some reluctance and did not act arbitrarily. Additionally, he was quite old at the beginning of his reign and spent the latter half too infirm to directly rule, much of the real power existed in the military officers and common soldiers who supported him.

Accusations of tyranny and comparisons to Julius Caesar were common and obvious ways to criticise the new government, especially by the Stuart camp in exile. Cromwell was able to managed this dissent through a robust secret service led by John Thurloe and the military influence of his "Grandees" (senior army officers). However, Royalist sentiment in the Commonwealth was not always negative towards Cromwell. Many members of parliament felt that the only way forward for the government was to crown Cromwell as King Oliver I. They felt this way for a number of reasons:

1. Many people in England still couldn't perceive a permanent government without a King. This made the Commonwealth feel like a temporary state for the country without a real future after Cromwell died or the army disbanded. Most common people did not care for the return of the Stuarts along with more war, so some MPs felt that Cromwell was the best option for King.
2. Parliament existed in the shadow of the army. A great deal of the nation's budget went to paying the standing army which resisted any attempts by parliament to reduce this spending or to disband any part of it. Some MPs believed that crowning Cromwell would distance him and his family from the military faction and give more power to parliament.
3. Making Cromwell King would have placed his role in government within a more traditional and transparent context. As Lord Protector, there was little he couldn't do and parliament was constantly uncertain about what it could do. As Cromwell's age and health became a factor, parliament wanted clear devolution of his powers, especially in the event of his death. A constitutional monarch would have set a clear balance between Cromwell and parliament, this political system was what most English people were comfortable with anyway.

The Humble Petition of 1657 pitched the idea directly to Cromwell in parliament and, despite spending some time considering it, he inevitably rejected the crown. To placate parliament, he accepted a coronation to the position he already held, Lord Protector, and nominated a successor. However, he refused to make this position a permanent, hereditary title and potentially did so for the same reason he rejected the crown. Cromwell simply believed that it was God's will that a monarch was no longer necessary to rule England and the Commonwealth, neither was a permanent role of Lord Protector. Unfortunately, that made him a radical minority even within his own faction. Cromwell was also aware that accepting the crown would have alienated his army which was still instrumental in holding the Commonwealth together. It is possible that Cromwell realised that by accepting a crown that he had opposed for decades, he would be compromising his personal republican and puritan beliefs while also giving substance to his many detractors who labelled him as a hypocritical tyrant.

These decisions robbed his successor at any chance of legitimacy after his death. Despite technically being a hereditary successor, Richard Cromwell's title was not. Without the same support from the military and a strong precedent for his authority, Richard could not wield the same control over rebellious parliaments and senior army officers who either saw themselves as superior successors or looked to the Stuart restoration as a better career move. Richard was neither a monarch nor a military dictator so failed to exude the legitimacy of the former or the strict control of the latter. Cromwell can be blamed for this situation but it is possible that his reluctance to rule made him equally reluctant to establish a political/royal dynasty to succeed him and usurp royal authority.