Yes. But when "Lorentz invariance isn't automatic from the form of the expression" it does NOT follow that you don't have Lorentz invariance.

Of course. Did part of what I said suggest I thought otherwise?

I guess the part about the “when you quantize it after fixing the gauge in a way that loses the manifestness of the Lorentz invariance, if you aren’t in the critical dimension, supposedly you don’t keep the Lorentz invariance” part could imply otherwise? If that part is wrong, my mistake, I shouldn’t have trusted the source I was reading for that part.

I was viewing that part as being part of how you could be right about Lorentz invariance being something derived nontrivially from the theory.

Because, the Polyakov action (and the Nambu-Goto action) are, AIUI, typically initially(at the start of the definition of the theory) formulated in a way that is not just Lorentz invariant, but manifestly Lorentz invariant,

and if there is no step in the process of defining the theory that isn’t manifestly Lorentz invariant, I would think that Lorentz invariance wouldn’t be a nontrivial implication, but something baked into the definition throughout,

so, for it to be a nontrivial implication of the theory, at some point after the definition of the classical action, something has to be done that, while it doesn’t break Lorentz invariance, it “could” do so, in the sense that showing that it doesn’t is non-trivial.

And, I was thinking this would start with the choice of gauge making it no longer manifestly Lorentz invariant.

I trust you have much more knowledge of string theory than I do, so I would appreciate any correction you might have.