This sounds about right. The antenna that works your remote start is on your rims or around them. So you can not change your rims to aftermarket unless you move the antenna. If you rotate your tires you need to make sure you get the antenna and move it to the correct spot or well it will not work. I think this service adviser has figured out the key to make good air into pure S)9t.
I know for a remote start they put the antenna up near the rear view mirror to make sure they have the range. I think the OEM antenna is under the hood by the firewall.
Take the key fob to autozone or a parts house and have it checked. They have a little device that you point it at and it will let you know if the battery is good or bad. Which tells you the fob is working.
1968 Coupe <53000 miles on her 289 came off the line Jan 21st 1968. I bought it in 1983
2013 Shelby GT500 Born 12/06/2012 number 439 of 530 in DIB and 1 of 9 with Red Stripes over the top and on the seats