The Parliament, and specifically the elected Commons, get to make the rules. If Blair didn't have a majority for even that decision to avoid debate he couldn't have done it †
This is why Theresa May couldn't get anything done, she didn't really have a majority for any actual policy. She needed the DUP (barely "allies" in any useful sense, even when substantially bribed to stay on side) and so even small rebellions in her own ranks meant she was constantly in danger of No Confidence.
It's uncomfortable to know that the majority of the people you sent, some of whom may have assured you personally that they agree with you, actually couldn't give a shit and are focused primarily on their own careers. But there it is.
† The Government controls the Crown powers, but Parliament's confidence in the Government is needed, any majority against the Government can get rid of it, with a single motion, "That This House Has No Confidence In Her Majesty's Government". This was not a realistic threat against Blair, or, sadly Boris, because for all the pantomimed outrage they're actually annoyingly popular - but it is why May couldn't get anything done.
Yeah, there was even a bizarre period prior to the last general election when the majority of the House of Commons was in fact opposed to Boris but didn't want to hold a confidence vote because they knew he'd win the resulting election (and couldn't find a majority of MPs who'd support any replacement). So instead they tried to puppet him around using their legislative powers whilst carefully avoiding the possibility of an election, which really isn't supposed to happen. All this was only possible due to the ill-advised Fixed Term Parliament Act, which made it impossible for the Prime Minister to force an election or make any vote on government policy into a confidence vote. This failed in its goal of stopping governments calling an early election in the hope of electoral gain, but did block an election when the current prime minister had lost the confidence of the house and the UK parliamentary system really needed one. The press avoided pointing out how much of a disastrous anti-democratic mess this whole thing was because it was helping the anti-Brexit side...
In order to form a British government you need a working majority in the House of Commons (the elected half of Parliament), this is because you need Confidence, as if a majority don't have Confidence in the government it can be dissolved.
The very least you can scrape by with in practice is called "Confidence and supply". An agreement between your political party and whatever other tiny parties or individual members can make up a simple majority (half plus one vote) of the Commons, to vote that they have Confidence in the government, and to vote through "Supply" bills, taxes and spending.
Parliament can choose to do whatever it wants, but as we see, in general the Parliament will be dominated by the governing party. So to an extent this is a distinction which makes no difference. Except, as Democrats are keenly aware in the US, just because somebody is a member of the Party doesn't magically mean they do whatever the executive led by that party says they should do...
So yes, in practice the UK Government and Parliament are much more cohesive than in the US where the Executive and Congress are run by completely different people with different agendas even on the rare occasions they're politically aligned, this is on purpose, but not by definition. If Johnson annoys enough "back bench" (ie elected politicians who aren't part of his executive) Tory MPs there's nothing he can do to ensure they vote how he wants and then he's probably fucked.
This is why Theresa May couldn't get anything done, she didn't really have a majority for any actual policy. She needed the DUP (barely "allies" in any useful sense, even when substantially bribed to stay on side) and so even small rebellions in her own ranks meant she was constantly in danger of No Confidence.
It's uncomfortable to know that the majority of the people you sent, some of whom may have assured you personally that they agree with you, actually couldn't give a shit and are focused primarily on their own careers. But there it is.
† The Government controls the Crown powers, but Parliament's confidence in the Government is needed, any majority against the Government can get rid of it, with a single motion, "That This House Has No Confidence In Her Majesty's Government". This was not a realistic threat against Blair, or, sadly Boris, because for all the pantomimed outrage they're actually annoyingly popular - but it is why May couldn't get anything done.