I'd still have to search through the list to locate that one train so the most time-consuming task would remain the same. However, I ended up finding some sort of workaround for that part.
https://drive.google.com/file/d/1n6i1pEvQtByqLk9hLpBrRdK8Fe3tlLYS/view
https://drive.google.com/file/d/1Oaqz65v2F6SvZFU59HRZSVLFUpYhj1kQ/view
It appears that no matter what temporary stations are added, even though it's before the standard ones, the list will remain sorted accordingly to the standard stations. In other words, as long as you use some sort of special character to have your personal train's station appear at the beginning of the trains list, it will remain so.
So, thanks to this… uh… discovery (I have some sort of "late to the party" feeling right now…) the issue is basically solved since finding that personal train is now just a single click away.
If for any reason you happen to still want to add that feature though, there is a mod that could give you some inspiration: https://mods.factorio.com/mod/autodrive
Dorfl added such a function in his latest update - and that's actually the reason I thought about it for Temporary Stations.