I actually think that's a comparatively minor detail, and not all that important here. The biggest issue that has often been discussed is the first past the post constituencies/districts/states both systems share. Neither Trump nor Boris would have become President or PM without that, simply on account of voters having more options to cast a meaningful vote for (UK also does a little bit better than the US in this regard).

it's the important detail

in the UK the electorate don't pick the leader

the electorate vote for an MP, and the group of those that form the party alone select the leader (and they can dismiss them just as easily)

the party has power over the leader, which is absolutely not true in most presidential systems

(and as to your second point, yes, under some hypothetical system that doesn't exist it's possible things may be different)