Had the macOS 10.15 Catalina HL-2140 driver (https://support.brother.com/g/b/downloadlist.aspx?c=us&lang=en&prod=hl2140_all&os=10060) been tested? Since this is not a supported macOS Sonoma 14 driver, it is quite possible that it does not work, especially not via USB.
I guess that you have tried to reset the printing system and to re-add the printer:
Reset the printing system on your Mac to solve a problem – Apple Support (UK)
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)
If nothing else helps, you may have to look for another solution or workaround. Perhaps, a special print server in order to allow an otherwise unsupported printer to be used with a modern Mac (or iPhone/iPad) in an AirPrint-like way.
Techguyuk mentioned a Raspberry Pi 3 in connection with a Kyocera laser printer.
Can’t print with iPadOS17 - Apple Community
Tesserax reported about using a Raspberry Pi 3 Model B+ for an old HP LaserJet printer.
Can I connect my laser printer to my netw… - Apple Community