I found this:
https://forum.no-intro.org/viewtopic.php?f=6&t=4239It appears the Famicom Mini / NES Classics uses 1MB ROM chips and mirrors it over the whole ROM space (except for the FDS games which do use 4MB ROM chips). Dumping the whole ROM space results in 4 copies of the same 1MB ROM.
If the Everdrive doesn't mirror the 1MB ROM 3 times, it will probably not work correctly if the games sometimes accesses the mirrors to confuse emulators. This would work on the 4MB versions because the ROM "mirrors" are there (though they are physical parts of the ROM and not really mirrors in this case).
The anti-piracy measures are described here:
https://mgba.io/2014/12/28/classic-nes/It mentions mirroring, but only for RAM not ROM. Most of the tricks described here are probably only tripping up inaccurate emulators and not flashcarts, but "Trick #4: Save type masquerading" might make it so saving doesn't work on flashcarts.
My guess is that these games relies on the ROM mirroring to work correctly and the Everdrive doesn't setup the mirroring correctly for these games.