I have had genuine friendships and other close relationships with introverts. I know how that plays out and that can work. Problems arise when people get weird ideas like because they talk to me once in a while, they have some special claim on me and I am not supposed to have other friends.

Keep in mind that it takes two to tango, even platonically, and if you like dancing with this person but they like dancing a lot more than you do, it is not reasonable to expect them to sit on the sidelines waiting for you. Let them dance with other people in peace at times when you don't want to dance anyway. It isn't taking anything away from you if they are consistent about being available for you when you need a friend.

Ultimately if/when some other people don't like you, well, that doesn't define who you are as a person or that you need to be fixed. It's just not the right fit between you, and that's nobody's fault. It's going to happen, and it only matters as much as you decide it matters. It really doesn't need to matter!

Be yourself, focus less on your perceived shortcomings/insecurities, and focus more on improving other people's day.

Unlikely. After all, they'd expect you to act the same way as them -- friendly despite not being a friend.