Dell E525W not recognizing (non Dell) toner and grinding gear problems

You should have never updated the firmware - this is a rebranded Lexmark! Lexmark (but they make up for it by only playing games with non-re-chipped LRP toner and let you reuse regular chipped toner for refilling). Epson just blocks these by blacklisting the chip once it's marked as "low" on ANY Epson and stores the chip ID in memory to recall the chip data to resist resetting.
That said, the rebadged Brothers aren't prone to it like the Lexmark models... They don't pull this crap as badly as Epson does BUT THEY DO BLOCK b/w printing when you're out of color on color printers with no override to force it into B/W mode which Lexmark officially added when they released the GO line. I consider this flaw fatal.
*Unlike Epson's blanket blocking, Lexmark just blocks reused LRP (Return program) toner from being reused without a re-chip. Regular toner can be refilled and remanufactured all day, but if it's re-chipped, it will need to be overridden. The problem is while the Lexmark variants are known to have it, Dell may have removed it or you need to try some "legacy" override methods from the period Lexmark used and Dell didn't mention it can be done; look at some ~2015 Lexmark manuals.

I have no hesitation whatsoever in sending the old ones which lack the override ("continuable stop") to die; I won't use them unless there's something REALLY big in its favor like a low PC and good toner; if it's average or bordering on 100k pages, the thing can die in a scrap bin. These will shutdown when any color is out and refuse to print black only. The GO era gets a pass from this since they all have had it to date when I look (B3442 mono/C3326/C3426). This "Dell" is most likely based on an old variable Lexmark :-(.

Your best bet is to downgrade the firmware if you can find an old version online with a matching checksum and see if it works. With the 3rd party "FW bomb" Epsons you can sometimes find old firmware and how to get into recovery mode on the control panel, so it’s worth a shot to try the firmware before you re-chip the yellow toner.
IF you cannot downgrade due to a downgrade block being in place which is common with Lexmark, you will need to do it in service mode and potentially need a "service mode" firmware file (or recovery image).

Now for a 2nd workaround: It may be possible to re-chip the yellow cart as well, as another potential workaround; BUT you need the third-party toner override command to do it; as long as the hex data can say LEXMARK or DELL without issue or Dell doesn't try and uses the same LEXMARK coded chip; if it rejects the LEXMARK hex (unless you find a clone DELL hex chip; they use the same carts), no bueno. Dell will not help you with this for obvious reasons. Apple did this in the 90s with Canon printers, but they just put a new label on Canon carts. People caught on and the lack of options to block swapping by hex code (effectively) meant that people found the Canon cart number and put reman Canon carts in these Apple printers for decades even when Apple left after Jobs returned. HOPEFULLY they did the same move as 90's Apple did because of laziness.
NOTE: You will need to subtract your current usage if you chip swap the toner.

Dell is in the computer business; they sell printers because of companies who want their printer from the company they buy their computers from and extra revenue. The R&D doesn't make sense for them, so they contract it out to Lexmark or Brother - the chassis and toner style gives you a clue. This works well for us, but also puts us at a disadvantage on these Lexmarks since Dell inherits their 3rd party supply FW blocks by default even if it doesn’t bother them as much, if at all.

DO NOT TELL DELL YOU WANT TO TRY THIS. THEY WILL USE IT AGAINST YOU IN THE FUTURE!