There have been attempts to do this, for example France had a postal TGV service for many years until decreasing letter volumes put it out of service.
There have also been proposals to carry air-freight containers on high-speed trains. There is no technical reason this can't work. But I guess economically it's difficult to make it pay. Otherwise I am sure somebody would have done it by now.
One of the problems is that air freight has become increasingly affordable over the years. Furthermore it represents a worldwide distribution system whereas trains would always be specific to certain routes. You would furthermore be competing in a sector that is at the same time too price sensitive to use air freight but also too high value to use normal intermodal trains. I think that sector is probably fairly constrained.
So, the constraint is probably cost more than pure feasibility with equipment. There's probably a maximum speed you could "safely" do with, say, open double-stack containers operations, and the added throughput from double-stacking likely outweighs any other benefits. Santa Fe ran the "Super C" as an express freight train back in the 70s, but it eventually faltered due to a lack of demand.
If you wanted to do this you would need to rework a lot of lines. Blowing along at, say, 125 MPH from Chicago to Denver is fine, but you'll lose a lot of that advantage in the mountains without expending a
lot of dynamite or doing a
lot of boring.
Also, don't forget the different weight, acceleration, etc. profiles of those freight trains, which would probably mess with any attempt to integrate them into ongoing passenger operations. The acceleration profile of a TGV is not going to be the same as a 50-car double-stack container train.
Having said all of that, the best way for something like this to become feasible would be serious restrictions on air freight as a matter of policy. If you functionally banned domestic/continental air freight (say, via heavy taxation) outside of certain narrow exceptions, then yes UPS, FedEx, Amazon, and the USPS might pool resources and pony up for fast(er) freight so as to preserve 2-3 day delivery anywhere in CONUS save for isolated corners. Going via Amtrak routes, NYC-CHI-LAX runs about 3215 miles (using the LSL and SWC). If you could maintain an average speed of 67 MPH, you can do that in 48 hours. Dropping that to 42 hours (two hours from pickup to station, two hours for switching, and two hours for station to delivery) would require averaging about 77 MPH. And if you push the average speed to 85 MPH, that gets you down to about 38 hours (allowing a larger distribution radius or some intermediate "swapping" stops).
The problem is that for something like this to become viable, you'd probably need something like $1tn in investment (since you'd need a network of 125 MPH+ tracks, probably mostly on greenfield alignments, plus the trains themselves and in-station infrastructure to do quick swaps of mail/package loads). The trains you might be looking at would be something like "X cars of passengers, Y cars of mail/light freight" (something like 10-12 and 4-6 comes to mind), and you'd need a
lot of them to enable efficient switching (since outside of
maybe some big hubs, you'd be loading/unloading car-loads instead of pallet-loads).
[FWIW, if the policy decision was made to get rid of almost all air freight,
