Still better to do that same thing (cert based auth) at the application layer instead of the network layer.
802.1x certificate-based authentication at layer 2 is a good defense in depth strategy.
I guess if you want to tie the network layer authentication to the user make it 802.1x with a user certificate. Either way you know the request is coming from authenticated endpoint and the user still authenticates to the application independent of the network layer. In all cases HTTPS protects the traffic from MiTM end-to-end.
Edit: oh wait, you mean have the applications check the certificate? Yes, but then you need support from the application. Does your printer do that, for example? You need to make sure everything does. You can of course do both.