Extracurricular math, or math enrichment programs often do introduce proofs in the realm of number theory. At the earliest level, it can include direct proofs such as "even + even is always even", "odd + odd is always even", etc.
The geometry proofs are usually taught as very structured -- a build-up from definitions and earlier theorems to the result, justifying each step, usually formatting the steps on the page in a particular way. There are many examples of interesting statements to prove, at a similar level of difficulty. Plus there is a benefit to visualizing things.
I think the answer to your question is that beyond very simple statements, number theory is not as simple an environment as it might seem:
Some statements in number theory are reasonably easy to prove (e.g. that $3$ divides $m^3 - m$), but it's not so easy to structure these arguments as a sequence of simple steps.
For many statements, to prove them rigorously, you need a proof by induction. But that's certainly a higher level of difficulty.
I think that's why number theory proofs are more commonly seen in advanced math programs, rather than in standard curriculum.