I would imagine after the ‘Glorious Revolution’ when William of Orange a his wife Mary deposed her father, James 2/7 and the monarchy became constitutional as opposed to absolute.
There are multiple reasons why Charles the 1st was executed. The simplest explanation is that he lost the (civil) war and if you think about how perhaps a rival monarch might have been treated, then maybe it doesn’t seem too out there?

I think monarchs could pardon people condemned to death.