Sources: The Revolutions podcast by Mike Duncan.
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 :)
Thanks!
He wasn't widely popular outside the parliamentarians, who were the ones in power. The "godly", who made up the majority of the Roundheads were reviled by a substantial portion of the British. Cromwell held the Commonwealth together, that's for sure, but the cohesion amongst the Roundheads dried up once he died.
Source: The English and their History by Robert Tombs
Of course the great irony is that he then arbitrarily dismissed Parliament himself (though the record doesn't suggest this was his actual goal) and was so unpopular in power that the restoration of the previous king's son to the throne on his death was inevitable. He certainly wasn't the poster boy for the development of British democracy, but he influenced its development massively - positively and negatively.
There is a decent chance that Parliament would have overthrown and executed Charles I without his military skill or enthusiasm for regicide (just like similar massacres in Ireland would likely have taken place without his -disputed- involvement and apologia) as he wasn't even a significant figure at the beginning of the conflict, but those whose name and public statements becomes most known always get the credit and the blame.