They go so far as to execute the king (that isn't something that seemed likely had by some strange reason the American's captured the king (some good fan fiction there)) and yet as far as I can tell really don't seem to have much of a plan / do much to establish much at all after that, and sort of roll over at the end.
It always seems very muddy with all sorts of religious motivations (that admittedly I find hard to understand) from the outside.
Then again the tradition of 'Parliament' (speaking in a larger historical sense) seems to swing wildly as far as its loyalty to the king, or lack of it, power, or lack of power, and so forth.
The religious motivation is fairly simple if you consider the period slightly before: Spain was predominant, England had gone to war with Spain, England was relatively weak, and Catholics had just tried to blow up Parliament...yes, England had lots of Catholics near government (which ended up being part of the problem) but it was a Protestant nation (the Thirty Years War happened in Europe too, this kind of religious conflict was not unknown). The extreme position that Cromwell took was also probably not a reflection of public opinion either btw (he was unquestionably not the leader of some democratic cause, and he just stepped into a power vacuum with leadership that was lacking elsewhere...the majority of his notable actions would be called religious genocide today).
To see the religious issue more clearly btw: if England had indeed had a Catholic ruler, it would have been like the Spanish Netherlands...that is the fear that existed at the time.
I think you're overestimating the average US citizen's knowledge of this and how it relates historically and politically to the area. I say this as someone who just had to look up what the Spanish Netherlands was, because at first I thought you were just talking about a hypothetical, not an actual region that existed.
"Throughout the early 1650’s Cromwell’s troops practiced a policy of scorched earth in Ireland which lead to the death by starvation and other causes of at least 200,000 and possibly as many as 600,000 in a total population of 2 million."
https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=118...
An alternative viewpoint to it being about religion is this [1].
[1] https://www.marxists.org/archive/hill-christopher/english-re...
But I still find it confusing even if they didn't have a sort of age of enlightenment like spin in the sense that ... they really didn't seem to have a good plan and it just sort of ends abruptly.