"[...] if programming and starting the FPGA part means a power cycle of the whole computer."
Why should that be necessary? The whole point of "soft" FPGAs is the fact that you can load - and reload - them with a bit stream from any source. A power-on ROM device is a limited way to use an FPGAs versatility.
Back in the 1980s my PC ISA prototype cards were populated with early Xilinx FPGAs. The cards interfaced as a peripheral address for both loading and passing data. The DOS program sent the download as a bit stream. The application reprogrammed the FPGAs' functions as needed - by generating its own bit maps on the fly in hundreds of permutations.
FPGAs nowadays apparently allow you to do partial reloads of a running chip.
It's a surprise that it has taken 30 years for the Holy Grail of FPGA program assists to achieve their potential in this way.
I really should donate my prototype card to a museum. Clocking at 100ns it was about the limit for hand wire-wrapping. Worked first time too.