Bitcoin would be be even worse off: The network itself is highly public and there is only one network... so you'd simply start one Bitcoin node to enumerate all the other publicly available ones. These attacks can be resisted— see the tor bridges arms race for an example— but it's better to let the experts in that area handle that for us and take advantage of our common needs. Bitcoin is very tor compatible, its a good mix.
I'd still like to see several somebodies who know a lot more about networking than I do work on transmitting Bitcoin traffic over different networks (along with bridge nodes to shuffle traffic between the network we have now and the new networks).I'd sleep easier if I knew that an as-yet-undiscovered bug in the network protocol we have now couldn't bring the entire system down. I'm confident we'd quickly fix whatever the problem is and I'm sure it would be back up and running within 24 hours, but it would be better if big merchants and miners and services could run two or more completely different bitcoin-network-stacks so they're less likely to be taken down by DoS attacks, bugs, or ISPs deploying deep packet inspection to try to block Bitcoin traffic.