Having trouble removing tap aerator/regulator

Try grabbing it with a pair of Channeloks,vise grips or regular pliers. Don't worry about scratching it up since you're replacing it. You'll want to turn it counterclockwise, or clockwise if you're looking down from the top, to remove it.

Id recommend these. I find them to be quicker and easier to use than normal locking vise grips and they have a better design for trying to grip on narrow rounded shafts like you have. The jaws dont have the play normal vice grips do which cause them to slide off. These also work like a pipe wrench/monkey wrench, such that as you squeeze and turn they will bite into the surface and not slip. They work great for normal nuts/bolts as well as smooth round pipes with less chance of stripping than vice grips. They sell them individually or in packages of 2-3-4. The 2 pack is probably your best value if you don't do work requiring them much.

If for some reason you can't unthread it then...replacement spigots/spouts are cheap. Not sure what plumbing issues you mention but the entire piece should thread off if you decide to replace the entire spigot(what i call where the water comes out). Just grab the entire shiny party with the same tool and turn it counter clockwise. They just turn onto a threaded nippe. It may unscrew at either end. If it unscrews at the far end then remove the nipple from the spout afterwards. Wrap threads with teflon tape before threading back in. (I hate pipe dope)

Some, which i have never ran into around here since most houses are literally 100 years old, may be held on instead by a set screw that just requires loosening. On the bottom side near the wall.

It shouldn't be, but if someone was super cheap and it is all plastic and flimsy (you did mention a plumbing issue) it could be one meant for an entirely different application. But i doubt this, though I've seen other versions of this hack.

https://www.lowes.com/pd/IRWIN-VISE-GRIP-GrooveLock-2-Pack-Tongue-Groove-Plier-Set/1001525148