Disk Space critically low, now I cannot log in.

Try using Target Disk Mode. You will need a suitable connecting cable. Ethernet won't work.

That will make your MacBook Pro appear to be an external hard disk drive that can be used as you would any other disk, including the ability to reclaim sufficient space for it to boot.

Getting past its Permissions might be an obstacle.

Target Disk Mode does not work over USB, only Thunderbolt (or Firewire for by now very old macs).

You need either a TB3 cable (which visually looks identical to USB-C but they are NOT interchangeable), if both macs have TB3; TB2 cable and, if one of the macs has TB3, Apple’s TB2-TB3 adapter.

There is a solution that doesn’t require another mac. It does require some Unix skills and that you have your important content backed up elsewhere. You need to start in Single User mode, mount your drive:

/sbin/mount -uw /

then use the Unix rm command to remove stuff. Don’t do any of this unless you are sure you know what you are doing, as doing it wrong risks losing data. You will be working as root with power to do nearly anything.

Yes. You wrote that you tried another Mac, and strictly speaking you don't need another MacBook. It can be any Mac, but it must be one that can read an APFS formatted volume (meaning, you need a Mac running High Sierra or later).

This was using a USB-C to USB-A cable. What am I doing wrong?

That cable won't work. The cables that will work are explained in How to use target disk mode to move files to another computer but that does not mean to exclude Thunderbolt to USB-C adapters that might work. If you need sources I'll try to find one.

Luis Sequeira1 posted another alternative. One more to consider is to extract the MacBook Pro's internal hard disk drive (see * below) and install it in an external USB hard disk drive enclosure. They're cheap. It would appear to another Mac the same way as in Target Disk Mode, if it worked.

(*) assuming it has one. You didn't provide your MacBook Pro's model designation. A MacBook as shown in your profile information is a different story. None of them have rotating hard disk drives. Your MBP might.