There have been test-stand burns of the main engine design, if memory serves. Boeing is building those. The SRBs are closely based on the STS’ SRBs.
The sad fact is, a vehicle using just the four SRBs for the first stage would duplicate, actually exceed, the capability of the finished system. The cryo system is along for the ride, just as it was (for the most part) with the Shuttle.
Putting a fifth SRB in the center in order to play the role of the second stage, pushing the remaining 100% cryo (much smaller, as they were in the Apollo launch system) would put us on the Moon as early as getting them constructed, no joke. The difference is, the ride would be more like Mercury than Apollo.
That’s some serious heavy lift.