My go-to guitar for musical theatre is a Line6 Variax. I own two, for variety as well as resilience. The only thing it can’t do is nylon-string, so I also have a Godin electro-classical and an Ibanez ukulele. With those three instruments, I could cover pretty much any show. But I love my (small-ish) collection of other guitars, and look out for the chance to use them for shows without too much variation in guitar flavour required.
The Variax is superb – I have a couple of JTVs, which do support custom tunings, as do the newer Standards. Don’t go for an older one (300, 600, 700 etc.) as they have less processor power and don’t support tunings as easily (I think it’s possible, but more complex). The bad news is that the Variax was discontinued last year… although that might mean you can pick one up for less. The Standard is probably the best value, although I don’t like how they look!
There aren’t that many shows that need different tunings, but there are one or two where you need Drop D, and Spring Awakening also has a song in DADGAD. I also fake a mandolin using custom tuning and a capo.
If you’re thinking about a Variax, then that’s extra reason to go with a Helix – you can connect them with a digital cable, which is silent and also lets you control parameters across devices. I haven’t done it much, but it’s so cool using the tone control on your guitar to adjust the delay time in the Helix. Or change a patch on the Helix and it changes the guitar model (and tuning) on the Variax!
MDs typically don’t care too much how we get the sounds. I think there was only one show where the composer was involved and he wanted it all on a real steel-string acoustic. And certainly most MDs prefer the practicality of not having big/loud amps in the pit!