So to take your example of real numbers—if someone didn’t want to use SOL, they would still prove the same theorems, they would just end up proving that they are true for any Archimedean complete totally ordered field. In general, I think most mathematics (i.e. mathematics outside set theory and logic) is robust with respect to foundations: rarely is it the case that a change in axioms makes a proof invalid, it just means you’re talking about something slightly different. The idea of the proof is still preserved.
So to take your example of real numbers—if someone didn’t want to use SOL, they would still prove the same theorems, they would just end up proving that they are true for any Archimedean complete totally ordered field. In general, I think most mathematics (i.e. mathematics outside set theory and logic) is robust with respect to foundations: rarely is it the case that a change in axioms makes a proof invalid, it just means you’re talking about something slightly different. The idea of the proof is still preserved.