I’m wondering why I can’t use a 125kHz chip? Also, if I do use a chip like this can I rewrite it?
I would suggest you can use whatever you want, however that would depend on what your use case scenario is.
If you only ever want to use 125kHz then that is your best option.
If you want to potentially future-proof yourself, I would recommend you potentially look at the NExT which gives you the added capability of 13.56 MHz and you get free shipping; whereby if you just get the xEM you have to pay the postage so if you were going to pay extra you might as well pay a little more to get the extra technology.
As far as writing to the chip, on 125 kHz that depends on what you have access to:-
Chinese cloner which is not particularly reliable but easy to use.
The much more expensive proxmark that IS reliable but not particularly easy to use.
If you have access to the system you’re trying to utilise you can just enroll it without needing to rewrite.