Livre d'Or

Yes, a lot of people like to frame the relationship between parasites and hosts as a kind of arms race. It's certainly true that viruses can replicate really really fast for exactly the reason you give. But it often isn't advantageous for the virus to just keep replicating, taking control of all the host's resources, until the host is nothing but a festering pile of virus. Because once that happens the virus usually has no way of spreading any further. Whereas if the viral replication rate is limited so that the host immune system can eventually overcome it, the sick but alive host is able to walk around infecting lots of other people. Or if the virus barely replicates at all, but lies dormant for many years, the host may infect hundreds or thousands of other people. (This is the case not just for viruses, but for other parasites too, it's often better for them if they can keep their host alive rather than spreading too dramatically and lethally.)

It's also true that animal cells have the ability to simply commit suicide if they detect a viral infection. So waiting for the immune system to respond, which I agree is slower than the fastest possible rate of virus replication, is not the only way for the host to get rid of the invaders. If the viruses really hijack the cell system too dramatically, there's this kind of dead man's switch system in place and the cell will just break up into pieces if it starts making viruses at the expense of survival factors.