The following list does not specifically mention macOS 11 Big Sur as a supported system (only macOS 12 Monterey and newer):
https://support.brother.com/g/b/oslist.aspx?c=as_ot&lang=en&prod=hll1230w_eu_as
However, if you take a look at the downloads page for macOS 12.x to mac OS 15.x, you will notice that Brother recommends using AirPrint.
https://support.brother.com/g/b/downloadlist.aspx?c=as_ot&lang=en&prod=hll1230w_eu_as&os=10072
The Brother HL-L1230W is also listed as a supported model in About AirPrint - Apple Support (en-us edition, January 13, 2025).
Normally, an AirPrint printer can be used without additional drivers by Mac operating systems as low as OS X 10.9 Mavericks or even Mac OS X 10.7 Lion, provided that the printer and the Mac computer are connected to the same network (Wi-Fi router).
The reason for not mentioning macOS 11 Big Sur as a supported system for this printer model is unknown to me. Could it possibly have something to do with added AirPrint USB support in the printer firmware for more modern systems? I do not know.
You could of course carry out a simple test with AirPrint and macOS 11 Big Sur anyway (without additional software). If so, use a wireless printer connection to a Wi-Fi router.
https://support.brother.com/g/b/faqend.aspx?c=as_ot&lang=en&prod=hll1230w_eu_as&faqid=faq00100735_000
By the way, which exact model was in use before, and do you still have that old printer?