Why are we starting runs when we press the continue button and not when the game loads in?

Additionally, if we use a program like Auto Split integrated with LiveSplit, couldn't we get rid of load time in the timer? Then everyones runs would be based only on active gameplay.

I can offer a few guesses as to why that was chosen as the start cue, but a mod would know better. It likely has to do with ease of verification, and is also an easy cue for the runner (simply press 'split' and A at the same time).

If you want to program or configure an auto splitter, people will appreciate your effort.

Typically, it's not just a "one click" setup — an individual or group of individuals needs to analyze raw footage from different runners/capture setups, build a database of loading frames, and do sessions of testing, tweaking, and debugging. In a visually rich game such as this, the utmost care will need to be taken to avoid false positives. It doesn't help that the in game load screens have multi-frame animations, either. Actually, the easiest aspect of this theoretical graphic auto splitter would be the stop time, since that is just when a specific cutscene starts, which should be very easy to work with.