How do you force your representatives to actually represent their constituents?
Petitions have a place, which is to inform of a point of view and of the opinion of a portion of the public. That's a form of lobbying. But that's it, we should certainly not expect that a law be repelled because of a petition, and rightly so.
I get your point about petitions and direct democracy being a form of who shouts louder (in the media, advertising, # of campaign events etc), but this is equally true of regular elections. It's even more so in a first-past-the-post system like the UK, whose two major parties have no interest in shifting to a proportional representation system because it would advantage smaller parties at their expense, even though the result would more closely reflect public preference.
In my view, parliamentary systems developed a few centuries ago have their advantages but also come with a great deal of historical baggage (systems that benefit a particular class of candidate and so forth), and they're buckling under the pressures of a real-time information society where people know transparency and timely publication of information are technically possible but such goods are systematically withheld from the public.
They instead propose a bill that will cut jobs
There's deliberation, but a lot of other people want to cut jobs
Is you shouting "hey, that is not what I voted for!" yelling and disrupting process, or calling out the fact that you were lied to and your representative is in fact not representing you?
MPs belong to political parties - consider what happens if an MP's constituents and an MP's party disagree?
They might be allowed to vote against the government, if their vote will have no effect on the bill's passage - but if they actually stop the bill's passage? They're kicked out of the party, which will make the next election extremely difficult for them.
MPs are elected for reasonably long terms - and that means they regularly do things that weren't in their party manifesto. Nobody running for election in 2024 had a manifesto policy about 2025's strikes on Iran, after all!
That flexibility means they can simply omit the unpopular policies during the election campaign. A party could run an election campaign saying they're going to introduce a national ID card, give everyone who drinks alcohol a hard time, cut benefits, raise taxes, raise university tuition, fail to deliver on any major infrastructure projects, have doctors go on strike, and so on.
Or they can simply not put those things in their manifesto, then do them anyway. It's 100% legal, the system doing what it does.
It's been decades since the UK had any genuine bottom-up policy representation for ordinary people.
Petitions are the only mechanism which produces some shadow of a memory of a that.
MPs are elected to Parliament, they get input from their constituants. Bills are debated, revised, voted on multiple times. There are consultations and input from a board range of view points.
A petition is in effect trying to shout over all that process from the street outside.