I have just described how the public drives the democratic process to ensure everyone gets a voice, not just whoever shouts the loudest. That's the opposite of ignoring the public.

If the public truly drove the democratic process we'd have proportional representation or something other than the current system.

That's the nice-sounding theory, but I don't see any metrics on how well it works in practice. MPs aren't required to share the input from the public or publish lists of how they voted on every issue prior to elections. Representative democracy really includes very little accountability for the legislators.

It certainly works better than to govern according to whoever shouts the louder.

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.

Incumbents in a bad system always argue that it's better than their worst characterization of the alternative. The reality is that elected officials still have very little accountability. They're only subject to re-election once every few years and it's virtually impossible to get rid of one mid-term unless they get themselves arrested.

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.