Switching to Stronglifts - muscle gain or just strength?

OP, look at the link I provided before and make the decision yourself. I think there are answers given in this thread that include correct and not-complete information is causing some confusion based on your beginning question of if the program adds muscle mass and additional comments you replied to of "As long as I gain muscle/strength and burn fat..."

Can you gain muscle in a caloric surplus, yes of course, this is a no-brainer.

Can you gain muscle in a caloric deficit, yes of course IF... see information provided by link to see if you fall into these scenarios. http://www.bodyrecomposition.com/fat-loss/adding-muscle-while-losing-fat-qa.html
If you do not fall into those scenarios, then the answer would be generally NO, but again read the article as its not a 100% absolute NO, just generally no.

Will this muscle gain (in a caloric deficit) be short lived, again if you fall within the parameters of the information given in link, YES.

Is it better to cut to lose weight and then bulk to gain muscle, YES absolutely in my opinion. Trying to do both at the same time (recomp) by say eating at or really close to maintenance is spinning your wheels...again, my opinion.

Stronglifts is a beginner powerlifter routine, it is not a bodybuilder routine. There are a good amount of fundamental differences between the two types.

It is geared towards developing strength not really maximizing size.

Don't use it and expect to look freaking awesome at the end. Use it to be able to lift awesome at the end.