That is still a fair amount of effort. The goal here is not the get rock solid proof, but to filter sufficient people that deportation costs remain low.

Just like motivation letters for a job: they show effort and serieusness. Even though it is trivial to let someone else make them (or let an AI write something). They still filter a decent amount of applicants.

Even if you managed to trick your way into staying longer, then what?

With limited time access your continued stay is illegal, and you are likely to go to jail and be deported the first time you need to prove your identity for virtually anything, cant, and the authorities get a hold of you. Or had to go to the hospital. Or tried to rent an apartment. Or even tried to make a cash withdrawal from your account.