Staleness of what though?

LSPs only really pro-actively send diagnostics (error/warning/info/suggest[/code action]).

Everything else is responsive; the client asks for symbols in this document, or completion on this line, etc. And if the client is aware of document changes (which are versioned), it should notify of those before requesting new symbols/etc, but that's not difficult.

I don't know that it's mandatory, but I definitely implemented servers so that they would complete processing changed documents before responding to any later requests.

And if it's just the client re-using cached symbols without asking for an update (which should be very fast if nothing has changed); well, that's foolish.