From what I can gather, it sounds like things will keep working until something changes on the back-end that leaves the old Steam client unable to connect anymore. I don't think they can't say when that will be exactly though because it depends on future updates.
The problem is when those "older builds" rely on a connection to a back-end. If this was just a standalone piece of software that is one thing, but you can't just let out-of-date clients that connect over the internet to run indefinitely.
You could likely set it up to work "offline" so you can still play your steam games.
If you were to set the steam client to offline (Assumedly through the Mac top bar 'Steam > Offline Mode') it should never need an update or contact with servers to keep working.
That said, I don't know if there is a limit on how long you can have a computer connected to a specific account while never connecting to Valve's servers.