What's wrong with client certificates? Instead of reinventing the wheel they should've just used those which would've given browser vendors a reason to improve their UX regarding client certs.
With hardware U2F the benefit lies in not having the private key available on the client device at all. That means that copying it is impossible (without dismantling the key and using quite advanced equipment to attempt to read the private key). With software U2F I think you are right; client-side certs just work, now, in all major browsers. Installing them is a hassle, but it can be managed with good documentation (…
[0]: https://developers.yubico.com/PIV/Introduction/PIV_attestati...
U2F is designed for only one algorithm and allows a lot of optimizations (e.g the private keys are not really stored on the device but rather generated from master seed and origin). That's why they are substantially cheaper than PIV devices.