I dunno, I'm torn on this one.
On the one hand, making someone take a taxi or shuttle from the airport is a huge dick move. Thats what people who dont know anyone in the city they just landed in have to do. If a friend or family member lands at your airport and you make them take a taxi, youre a dick.
On the other hand, it aint your friend, its your moms. I can kind of understand your mom asking you, honestly. Its a huge dick move to make a friend take a taxi from the airport. Your mom doesnt want to be a dick. The only uncool part is assuming youd do it. Asking you? Thats not even a little rude, its just asking a favor.
Both positions are understandable. The only part thats not understandable is this "oh poor me I'm an introvert and might have to talk to people" shit. Thats a weakness, not an excuse. Get over your weakness. On balance, my ruling is: you need to pick up the person at the airport to get over your insecurity. A grown man should be able to pass a couple hours in the company of a stranger without discomfort.