It's good to get the low hanging fruit and easy wins out of the way, whether cosmetic or electric or mechanical. That way you get some motivation to keep going by seeing some improvements right away. Chasing the easy stuff might also reveal other things to change your focus. My procedure is generally:

1) Vacuum and wipe-down. You might find all kinds of random parts, hints, and tells.
2) Inspect all wiring, once-over... just to get a sense of condition, potential hazards, and functional level
3) When game seems "safe" to turn on, do so. Run diagnostics, play games, take inventory of needs

That inventory is key, and why every situation is different! You might end up with a game that looks great but needs functional work. Or a game that plays wonderfully but looks like crap. Mostly likely, a little column A and a little B - depending on how picky you are.

So then you review your inventory, budget and goals, and decide where to begin. Can you fix Cosmetics without affecting Function, or vice versa? If so, pick whichever suits your fancy. However, if this mech or that board can't be accessed without exposing other flaws, and/or it makes sense to "add on and do this while you're in there", or "track down the root cause"... well, every situation is different. Maybe you have to disassemble a game so thoroughly to fix something cosmetic, that you use the downtime to fix other stuff. It really doesn't matter, so long as you work toward your goal.

I've bought a few games that were complete and total basket cases and absolutely dangerous to plug in (or even touch) so I essentially was starting from near zero in either case. On those I would work in tandem: on the electrical / mechs for a while until I'd get frustrated or at a temporary dead-end waiting on parts... when that happened I'd work on the cosmetics (while pondering the troubleshooting in the back of my mind). When the cosmetic stuff ran into its own delays (waiting on supplies or weather) it was back to electronics. Eventually it all comes together, and basket cases are among my favorite projects.