EADG. Always have, always will.
The tension on the higher gauge strings is more likely to put the lesser gauge strings out of tune. That said, it's a pretty small consideration once the strings and neck are settled in. So long as temperatures are relatively stable. But when putting on new strings or adjusting a set up, it's more important.
I do have a bit of quirk though, I tune to songs that are in G rather than using a tuner. I find G is ever to slightly easier for me. Probably because I've always used songs that are in G
I don't bother with a tuner unless I'm on stage. In which case I intend to tune up ASAP on arrival, once again at beginning of soundcheck and once again at end. But usually just end up tuning at soundcheck and checking just before we begin.
To be honest though, with my bass and my strings (I use heavy flats) they don't go out of tune so it's all a bit moot unless I'm setting up a bass or stringing, which is rare.