So, is there an easier way to completely disable the microphone?

First take it apart and determine what form factor the microphone is.

Is it a through hole microphone with two pins that are soldered to the underside like this? If so it's best to desolder it to prevent damaging the PCB. Use a soldering iron to heat up the pins and pull it out with pliers from the other side. If you don't have a soldering iron and don't want to buy one, I've also seen people using side cutters to cut off the solder joints and loosening the component enough for it to be pulled out through brute force without breaking the circuit board. If you can't or don't want to do either of those and don't mind risking the device, you could just yank it out with pliers like you're an old timey dentist pulling teeth, and hope that the pins break before the board does, might also help to twist it back and fourth repeatedly until the metal gets fatigued and break.

It might also be a surface mount microphone that, as the name suggests, is only soldered to the surface of the PCB. Might look something like this. These are pretty challenging for most people to desolder especially if it's close to other components, but they're small enough that the solder pads don't need a lot of force to break, so if you can get a good grip with pliers you should be able to just rip it off. Twist it until you feel the solder joints snap and then pull straight up. It doesn't really matter if you rip the pad off the circuit board since you don't plan on soldering anything else to it. But what you do need to be careful of is if you peel off more of the copper traces than just the pad, which can damage other components. If you do want to desolder it, touch the soldering iron to the metal casing which should hopefully heat up the entire component enough to melt all the solder joints, then pull it off with pliers. Just be careful not to touch any nearby components with the soldering iron.

Failing all those, you could also take a screwdriver or awl, put the pointy end on the microphone, and hammer it a few times to cave the metal casing in and hopefully crush the audio sensing parts. This will probably destroy the microphone, but less certain than removing it.

I've also heard recommendations about grounding the microphone connections after removal for extra privacy, mainly to prevent the traces from picking up EM waves, but I don't know how to reliably do that without breaking stuff so can't give any advice.

Does putting tape over it completely mute it?

No. Speech is surprisingly robust from an audio perspective and it's entirely possible for audio not recognizable to humans as speech to still be decoded by speech recognition AI. The thing is even if this works 90% of the time and makes the audio completely unusable, you can never prove if your case is in that 10% where there's juuust enough information for AI to detect. You also can't easily hear the actual output of the microphone so for all you know it might be fully intelligible and just muffled. If you're concerned about privacy to the point where you're asking how to remove a microphone, I doubt you'll accept that 10% chance anyway and removing the microphone entirely will save you a lot of anxiety.