I have no problem supplying a house with generator that does not have GFCI. Most likely those stationary generators do not have GFCI. The protection is no worse than what you get from the utility.
If you look only at the system starting from the transfer switch down, the GFCI equipped truck with EGC severed is pretty safe. But I listed several scenarios above that I would personally want to prevent, where some low levels of ill-informed or accidental usage on the system between transfer switch and truck will lead to problems. Someone could probably make some YouTube clickbait showing how those scenarios have a shock risk.
If I owned this truck -- if nobody touches the house / truck besides me, then EGC severed should have minimal risk and I have no problems with it. Maybe put some permanent tags on the ends of the extension cord:
- on the house side - saying not to use with power tools
- on the truck side - saying not to plug in any tools
Unfortunately this is not bulletproof because it requires people to read, and often code wants physical lockouts against unsafe situations because people don't read.