You don't need one crank arm on to remove the other crank arm. Else, you'd never get both crank arms off at the same time. Right now, you have a semi-loose spindle, stuck in the bottom bracket, with a crank arm attached, right?

1) From the attached crank arm, remove the cap, probably using a hex key. This is the cap that screws into the crank arm and covers the spindle end. You don't need the other crank arm installed for this.
2) Using a socket wrench, remove the bolt that holds the crank arm onto the spindle. You don't need the other crank arm installed for this.
3) Screw a crank removal tool into the crank, where the cap was. The cap threads should match your crank removal tool's threads. You don't need the other crank arm installed for this.
4) Using two wrenches, both on the crank removal tool, screw the center part of the tool against the spindle. The arm should come off. If it's sticky, whack the center part of the crank removal tool with a hammer - this should loosen things. Remove the arm. You don't need the other crank arm installed for this.
5) Now that the crank arm is out of the way, clean the internal splines on that cup, and use the correct bottom bracket wrench to unscrew it. You don't need the other crank arm installed for this.

If you noticed, there is one consistent fact: You don't need the other crank arm installed for any of this. The other crank arm is immaterial. Doesn't matter. Stop thinking about it. Remove the crank arm that's still on the spindle, using a crank removal tool. Then use the proper BB wrench to remove the cup.

When you are removing an arm, either side, of a square taper crank, it is entirely immaterial whether the crank is installed, or partially installed, or completely removed from the bike. The other arm doesn't matter at all. Stop talking about the other arm not being there. The Park Tool video shows how to remove the arm after you've removed the cap and the bolt: