If your DC has been hosted at parties for this boy (and presumably lots of his other friends too) for several years in the past and you've never reciprocated I can see why the mum might be miffed. You were away for DS's birthdays but you could have occasionally hosted a Halloween party or something. You surely realised you weren't pulling your weight on this front?
Nevertheless, I think leaving one child from a friendship group out is pretty mean. And even if he'd been invited to the party, signing your DC up for an activity and just expecting someone else to take them is CF behaviour.
Your attitude about the child potentially being better than your DC is petty and, I hope, just fueled by your upset over the party and annoyance at the entitlement with the lift. Your DC will have rivals all the time, ones he is friends with, learning to handle that with grace and use it to help you improve yourself is part of the character building that sport is often exalted for. you should be helping him learn this, not giving power to jealousies.
I would try and put the upset about the party behind you, OP. Think about the expectation for the lift as a completely separate matter. Does your DS enjoy having his friend come along too? Does it disrupt you at all? e.g. get in the way of doing other things before or after, take over time that used to be good bonding, cost you a lot in petrol, etc. If the actual lift is not something you want to do for you and your DS, then let the mother know you won't be able to keep doing it. But if it's really no trouble and your DS is happy, keep it up, at least for now. Things will change in Secondary and you can reconsider then.