It's cost savings in the NA sense. They skimp on some things but not others to give the appearance of being fiscally conservative. And sure, on Genesee Ave that could have been at grade, but along I-5 to serve those destinations elevation was basically the only option.
As for stations, these mirror the designs of the at-grade stations on the system, so "consistency" was likely a large influence. San Diego also runs more conservative for a large city, so cost savings are a big talking point amongst politicians.
The choice to grade separate was mostly necessitated by the terrain in this area. The freeway sits in a deep cut surrounded by lots of small canyons.
There is a CPUC rule that makes level boarding difficult (even if they did say ADA preempts its, TAs are still afraid to implement it). The main impediment is that Caltrain doesn't really seem to think it's important given the bare bones staff they assigned to the study and the epochal timeline.
If they're so concerned with fare evasion or simply want to continue to use humans as fare gates they should just buy some actual fare gates. It would certainly save people a lot of trouble but I'm not sure that's their objective.
Even if Metro wanted to relinquish the express lanes, Caltrans would still demand the freeway to be widened because of their misguided beliefs about freeway capacity. That's where most of that billion+ dollar estimate came from.
The biggest issue for them was the 10+ mile long single track in the median of I-10 which would require a pretty significant investment to double up (last estimate was north of a billion dollars).