How do you see the relationship, if you at all think that it's relevant, between flat unramified finite surjections among analytic spaces, and maps that are covering spaces? Are these really equivalent categories, or am I fundamentally misunderstanding something?

@nfdc23: You said that "etale cover" means "surjective local analytic isomorphism", but I'm somewhat confused about what you meant. I was assuming that etale is defined for locally ringed spaces as being flat and unramified, and that SGA1 uses this definition of etale when it talks about finite etale surjection of analytic spaces. Are you saying that the definition I had in mind has the interpretation of being a "surjective local isomorphism"? Or are you just pointing out that in SGA1 he's talking about finite etale surjections, and not just etale maps?

True. I guess there are two issues I'm still unclear about. One is it is somewhat unclear to me why if $Y\rightarrow X$ is a priori a topological cover, the resulting map of analytic spaces is etale. But I guess that's a somewhat easy technical proof... The second issue is -- does this hold for $X$ singular? In that situation your argument no longer holds...