> [...] skipping the probe entirely, and just having a USB receptacle on the board [...] long with some circuitry to make the other end talk USB3.1.
Practically, this probably ends up fairly similar to what I mentioned above, since "making the other end talk USB3" means receiving with the bits it sends and responding in a way it expects.
But, yes, in-situ / solder-down probes can definitely simplify things because it gives you the flexibility to do things like remove the existing termination (assuming the termination isn't in the receiver IC itself) and use the 1x Z0 soldered-in probe as the terminator.
I just probably wouldn't do this part:
> then have a front-end IC that is hopefully available and that is fast enough to do the sampling
This gets complicated fairly quickly. It sounds like you want both real-time sampling (so you can do something with the bits and "make the other end talk USB3") fairly high resolution (so can see something in the eye diagrams). That calls for a digitizer with like >=5 bits and a sample rate of >100 Gsps... Which basically doesn't exist, AFAIK. Oscilloscopes which sample that fast are interleaving multiple digitizers, and this isn't really something you can DIY.
However, if you just want eye diagrams, you can use ET sample techniques (running at a pedestrian ~10-100 ksps), together with something to synchronize the sample to the USB clock, to fill in the entire horizontal width of the eye diagram. There are several catches: First is that it takes a very large number of triggering events to fill in the diagram. Second, the 100 ksps sampler needs a very short capture window (on the order of picoseconds). Third is that you "miss" most bits in the data stream, so you don't capture any real USB data while making the eye diagram.
Fortunately, you can buy a used sampling oscilloscope which does this for you. It captures the samples, relates them to the USB clock, and places them in the correct place on the x-axis. You just need to construct probes to bring the signal into the scope. The 11800 I mentioned is one such example, but there are others, too.