Do not quit, but do start interviewing. Once you get a better offer in hand, present it to your boss. If they refuse to match (or outbid) for you, then take the competing offer.
I'm not sure if I agree with this approach. I would say ask for a raise without showing you have an offer from another company. If they don't then leave. I feel like it is awkward to show them that you have offer and then stay. But It could be just me.
Personally, I don't care if they feel awkward, nor do I care to be perceived as loyal in matters of business. The awkward exchange wouldn't need to happen if they ensured that they're employees felt taken care of. You shouldn't feel accountable for that.
True, you have a point. Personally, if I wasn't fairly compensated I would just leave. I can understand that there could be situations that changing jobs might not be straight forward.