They will continue to breed, expanding territory unless there are predators. When they come across a culvert (or bridge), they will block it, eventually overtopping roads and washing them out.
It is interesting that we see our infrastructure as 'valid' or correct, but the work these engineers (beavers) put in as a nuisance. In an ideal world, the benefits to local biodiversity and ecosystem health should be balanced with the cost of relocating roads when necessary.
My dad is a crazy old German retired machinist/tool&die maker. I think he enjoys the challenge. Not everyone would.
Maybe part of the problem is that humans in North America are largely not hunting beavers anymore?
We're very capable of being the predators that keep animal populations in check, and plenty of people do still hunt, but I don't think I've ever heard of anyone hunting beavers in my life
When I got my hunting license, part of the safety course was about trapping. The impression I got from that was that trapping is basically torturing animals to death and infinitely worse than shooting them with rifles.
I've heard the tail is particularly tasty and easy to skin and cook.
Just like our fellow humans. Godzilla to the rescue!