I don’t see how that could happen without causing a massive stampede away from USEF shows.
The participants in FEI competitions are a much smaller group. Many of the people who show in the lower levels at USEF shows, which comprise a huge chunk of the entries at most shows, are also likely to show at competitions that are not recognized by the USEF, whether they are breed shows like AQHA or local circuits or just schooling shows that are not recognized by anyone.
The USEF would probably lose a large number of participants if exhibitors had to make that choice.