Some kinds of cheating can be mitigated that way, but it can't really stop cheats which just play the game more optimally than the user is able to, using the same inputs and outputs that a legit player would use. Aim assistance in shooters, automatic parries in fighting games, economy-breaking levels of automation in MMOs, and so on.

There's also practical limits to how much data you can filter out in complex 3D games, both due to performance constraints, and because culling information too perfectly can cause things to pop into existence too late under real-world network latency. The effectiveness of ESP cheats can be reduced, but not eliminated in practice.

However, this only solves the cheat problem to a minimal extent. There is a lot of important data that players should not be directly aware of, but which is important for the game. For example, it is important for calculating sounds to know where enemies are nearby, even though you cannot see them, which makes wall hacks possible, etc.

This is not how it works, most games that take cheating seriously already have a gameserver where most of the gameplay logic happen.

Doing everything server side does prevent cheating.

That only solves half of the cheating problem - illegal inputs from clients.

The other half is much harder to solve. For a simple example - my client knows that there is an enemy player around a corner. It knows exactly where that player is, because that player is walking, and making noise. A cheats could allow the cheater to see his opponent's player's model through the wall.

For a more blatant example, consider cheats in a first-person shooter that just snap your aim to the nearest enemy's head. This involves zero violation of the game's logic, and also makes the game completely unplayable for everyone in a lobby.