I don't know what else to tell you.
If you followed this troubleshooting mentioned by
@Kubas_inko,
, by downloading those *.cia files using a bittorrent client like Deluge and installing them with FBI, your old 2DS (I presume) should have had its DS(i) mode fixed.
A way to test this out would have been to try installing and playing some DSiWare games or using another DS game cart. Once you can verify your old 2DS can in fact handle DS(i) titles (correct me if I'm wrong that your system happens to be n2DSXL), I would have recommended putting the YSMenu kernel on the flashcart's micro SD card to see if this changes anything.
If you have Luma3DS installed on the CTRNAND, you can try booting the 2DS without its SD card inserted and then try launching that R4iSDHC Dual-Core 2014 flashcart. This would eliminate guessing if the problem is coming from that SD card inserted in the 2DS (not the micro SD card inserted in the flashcart).
After exhausting those previous advice in fixing the DSi firmware, using DSiWare & DS cart games, and putting YSMenu kernel on the flashcart, maybe your 2014 model is blocked from the DS whitelist. You could try bypassing this with one of apache's launchers:
If nothing else works, maybe the 2DS game cart reader is busted. The only way to check is to borrow and play with a DS or 3DS game cart.
Last but not least would be to ditch using that troublesome cart (made by r4isdhc.com) on the 2DS and get the much better
R4i Gold 3DS Plus (made by r4ids.cn) if your budget allows it. If not, you'll have to settle with playing *.nds roms off the 2DS SD card using
TWiLight Menu++.