Playing without raiding

There's a lot more to Classic than just endgame raiding, although it must be admitted that raiding really is the core endgame of classic. This isn't an exhaustive list but here's everything I can think of off the top of my head which you might want to try instead:

The levelling experience itself is a huge part of the game and it's not a rush to get to max level. Take your time and enjoy the progression of your own character to the fullest. Then there are always alts to level up through different zones, and the other faction too.

Some professions (like blacksmithing for example) are really very involved and require you to travel across the world seeking rare plans/recipes, completing quests, specialising in a mastery etc. and offer a great deal of endgame content.

Getting rich is a rush and opens up a lot of other options like collecting rare pets, items, and mats. I know plenty of guys who just play the AH alone.

In late vanilla Blizzard introduced the 'advanced armor' sets now known as T0.5 which were obtainable through long and arduous quest lines with lots of dungeon group strategy and the sort.

As others have pointed out, what exactly is it that makes you not want to dabble in raiding? Is it because you don't have enough time or fear that you won't be able to contribute to the group? Or is it simply because you prefer to play alone?

So much this! And I would like to add something to point 2:
Not only are the professions heavily involved but also for people that don't raid very rewarding. Some of the better gear outside of raids can be crafted.

I know someone who is playing just to explore and make money. Kind of like a Marco Polo type...