> Are there just so many of them in the cytosol that through sheer numbers, there's enough random chance they'll just shuttle into place when the polymerase needs them?

Yes. Everything at that scale is very small, very close together and moving very, very fast.

This kind of thing is very common at the cellular level: have a receptor/channel/process/thing that only one precise 3d molecule can fit into - and then just wait.

As long as there's a process somewhere to make/acquire that molecule, then one will be along in a few nanoseconds or so, depending on the concentration. In the meantime the recipient will just wait.

In this way otherwise independent processes can regulate each other, and respond in a concentration dependant way to changing conditions without any central control.