That really depends on how things are being done. Tor involves making a temporary, random choice of relays, and periodically updating that choice. If all your transactions are sent through a randomly chosen chain of mixing services, you might have some kind of anonymity (but this is poorly defined to begin with, since mix-nets are supposed to enable anonymous communications rather than anonymous payments, and so it is not even clear that what applies to one still applies to the other).

"What I was thinking about instead was the 'linking' property - which leaks so much information - why wasn't that avoided at a system level?"

Like I said elsewhere, there is no specific problem definition for Bitcoin, so why bother to ask such a question? Really if you want a digital cash system without a central authority, you need to first define what that actually means; likewise if you want that hypothetical system to not have this "linking" property, whatever that actually means.