Why do we use zero infinitives with make, let, have?

Loss of markers like the infinitive to in subordinate clauses is a sign of increased grammaticalization of the construction. That means that the verbs get bleached of
some meaning and become semi-auxiliary. After that parts often get stuck together
like a Tinkertoy™ assembly left out in the rain.

In particular, there is a small set of small verbs used in this way, called "light verbs" in the literature, and that set includes make, let, and have, as well as get. The reason why there are so many constructions is because English lost almost all its inflections centuries ago and has had to make up for them by developing new syntax. This has been going on for centuries, with styles in syntax changing (read Robinson Crusoe and Artemus Ward to see how fast they change).

The result is that English has thousands of idiomatic constructions with special senses of special verbs, mostly unorganized and strung out hither and yon in the vocabulary. As opposed to the nice neat paradigms that allow free word order in inflected languages. These constructions are the hardest part of learning English (outside of spelling), because they're arbitrary and because there are so many of them.