I'm trying to find out why certain (parts of) characters were replaced with others while developing the simplified Chinese writing system.
I want to know why 见 replaces 見? As in 问题，問題.
According to the Wikipedia, 来自于行书与草书的楷书化.
[注 1] lists some examples, 许多偏旁如讠[訁]、饣[飠]、纟[糹]、钅[釒]、呙[咼]、见[見]、页[頁]、贝[貝]、车[車]、东[東]、门[門]、马[馬]等来自草书。
Some simplifications were based on popular cursive forms (草书) embodying graphic or phonetic simplifications of the traditional forms. (from the English version of the Wikipedia page)
Therefore, 见 replaces 見 because 见 is the cursive form of 見.
You're asking "why did a committee (委員會) make a specific decision?" The answer is "because it was a committee." There is no explanation, that's just what the committee decided. It's important to remember that simplified characters were devised by a committee.
In this case the committee went with "Preserving the basic outline or shape of the original character".
This might or might not be helpful: