No, the only requirement is the fat32 format.
That it should be slow is the most blatant nonsense, just like that you can only update the bootloader with a higher version, in two steps you can go back to any other version.

It is not entirely clear to me whether the original 1GB card works or not, if it does not cause any problems it is indeed very strange that every other SDHC card is rejected more or less.
I can use any SDHC card up to 32GB in my Rider v2 that I have been using since 2009.

I can imagine wear and tear on the SD slot, or contamination of the contacts, then cleaning could be the solution.
Try it with a different brand of card, it may have been a coincidence, but with another device I've had problems with a cheap possibly too cheap card and that was also refused.

I would start by opening the Rider and checking the inside for damage, loose soldering and or water damage. Although this device must be splash-proof, you should not always assume that it will always remain that way
Replacing the SD slot is also an option to ensure that the contacts actually make contact there.

Do you live far from our border?
It is only an hour's drive for me, testing one of your SDHC cards in the Rider of mine can already provide an answer about the card itself, would be more convenient with someone in your immediate vicinity, but if there is no other option, I will step on my cycle in good weather and at least I have a goal to drive to, if your information here on the forum is correct.

Please contact me via PM, or the aforementioned address