The Humble Petition was a small number of MPs offering him the Crown as a carrot to try to persuade him to tone down the military rule of his generals in the provinces and give power back to Parliament. Cromwell rejected it after some deliberation, not least because he believed some of his military allies would turn on him if he did. He filled a power vacuum, but the support he got from the military was conditional, and the Parliaments he convened didn't back his agendas despite his allies designing the franchises and vetting the candidates. Even Cromwell himself didn't think he was popular as opposed to powerful, famously remarking of a cheering crowd that "they would just as noisy if they were going to see me hanged". He stayed in power due to skill at bridging factions who hated each other and commanding armies, but failed to build a legacy or unite people behind a cause which meant the Restoration happened without significant opposition (and the public got to cheer the hanging of his exhumed corpse). After that, not not many words were written in his favour until Carlyle decided to make him a hero of history in the 19th century.
In the absence of a Lord Protector, we might have seen an attempt at an English Republic instead, and it is conceivable it could have lasted longer.
It is true that regicide was not Parliament's
original intent and was not pursued until after the Second Civil War. But Cromwell was prominent in making it happen when it did.
It's been over a decade since I studied Cromwell so you'll forgive me for not listing every source I read on the subject, but suffice to say it's more than one podcast :)