Don't know exactly for ID8, but olders use memory card to save your progress.

What do you mean by only one race? When you start, the first story has you race against drivers from Lake Akina. (Teamside) Once you race all these drivers, more courses are unlocked. Each progression gives you a new course and new set of drivers to race against.

Do you have a card reader connected?
Try checking the service menu settings to see if continues are disabled.

I visited Arcade Club in the UK a while back and they had disabled the card readers, I'm not sure if they had disabled continues too.
But we was only able to play one race and then it was game over, you couldn't progress any further than one race.

All that means is simply that without a card, you can't beat the game. However, with a card, you can beat the game but will cost more coins with Continue disabled in the Service Menu.

Isn't this essentially the case with all ID games where cards are basically required to make any meaningful progress?

I haven't done a lot with ID8 yet, but my impression of all others up to and including ID5 is that without a card reader your car isn't going to get any better and you have no chance of doing well as races get more difficult. That said, I never messed with any of these games for very long without a card reader, so I'm not entirely sure.

At any rate, playing these games in a home environment without a card reader (or simulated card reader) is very much a waste of time in my opinion.

I'll eventually release a card reader script that works with ID8.