No. I mean a republic. Certain things about the American systems would appeal - Checks and balances from the separation of powers - but the implementation of those things in a UK government would be very different due to culture and country size.
The USA
is a republic! With a dysfunctional political system - I can't see why anyone would want to adopt a similar setup. What you describe as "checks and balances" is rather successfully done here by the House of Lords. The American system just has inertia.
Head of state could be by executive committee rather than a single person (I mentioned presidential system as your original post infered a president taking up the queen's role).
I have not advocated a president and I don't know how you could have infered that from my post, which was deeply critical of the US system.. I replied to your post, which was in turn a reply to dejong ... who wasn't advocating a president either.
David Cameron is PM because he is the leader of his party, he wasn't voted as leader of the country directly.
Technically, no. In practice, yes. Go out on the street and ask 100 people who they voted for. Those that give you a name will - almost exclusively - name the party leader, not their local candidate. Try it.
Millions of people watched the leader's debates - how many of those went to a local hustings? Barely a fraction.
A directly elected executive would have a better mandate and be more representative of the will of the people.
We already have an elected executive committee - the Cabinet.
Electing it separately would have practical difficulties - presumably you would want the best and the brightest on the Executive, but many would not want to run the risk of failure to be elected to the smaller chamber compared to the safer confines of their constituency. To get round this, presumably you'd elect the committee in a separate election, allowing MPs to stand with the assumption that they'd stand down as MPs in they win.
However, staggering the two chambers leads to often having opposing majorities in the two chambers (as often happens in the US), which results in them thwarting each other's attempts to pass bills and leading to inertia and stagnation.
Other countries avoid this by having a greater number of parties, but British politics is dominated by a 2-party system, just like the US, and would suffer the same problems with two elected chambers elected at different times.
It would also, theoretically, allow us to break from party politics - an independent presidential candidate in the UK wouldn't need heavy party funding for a campaign so you'd unlikely get the 2 candidate system of the USA and instead, more independents.
Oh, we're back to a lone President now? (S)he will still need deep pockets - look at Presidential elections around the world (not just the US) - these are major undertakings. The candidates nearly always have a large political movement behind them and financial backing.
The head of state doesn't have to be the head of a party.
No, but it's very rare that (s)he isn't. How many apolitical elected Heads of State can you name?
You talk of the stalemate in the budget (the negative), but the system of separation of powers has done an awful lot of good. There's unfavourable policies in the UK that could've been stopped or ammended for the better, if we were a republic. The stalemate can serve to get the best possible policies from government, rather than having flip flop right/left wing politics from election to election.
They might be "unfavourable" to you, but they are policies of a majority government returned by the electoral mandate from a plebiscite. To seek to frustrate that is to frustrate democracy itself. Why bother holding elections if the system is designed to frustrate any attempts to pass bills? Parliament should not just be a vehicle of establishment to preserve the
status quo - it must be
capable of (radical) reform or else we might as well close the HoP and just leave government to the Civil Service.