Client-side password stretching - danger of the salt being public?
Though the salt is not considered a real "secret", is having the salt
  that public a serious consideration for rejecting this method (i.e.
  client-side password stretching)?


Yes, it is better if the salt is kept secret, but the attacker still has to perform a full-on offline brute-force search even if they know the salt before a breach, it's just that this search can be carried out before a breach and then after a breach only a logarithmic effort per entry is needed for the lookups. And if they don't learn the salt, the "timer" at which brute-forcing starts is pushed back to the point where the database leaks.

Now this exact attack, the ability to pre-compute real-salt-based password hashes before an actual breach is part of the reason why OPAQUE, the current state-of-the-art asymmetric password-authenticated key exchange (aPAKE) exists, as no previous aPAKE defended against this kind of attack.

So TL;DR: Hiding the salt is better, but actually doing so may be considered too much work and it's not the end of the world if the salt leaks.