I've bought most of my own gear over time. One reason is because I don't trust rental service from places/people I don't know. I do know and trust the people at my local dive store, so if I rent anything, it will be from them. (I do, in fact, still rent a regulator setup from them, but that's next on my purchase list.) My second reason for buying my own gear was so I could get something that fit me specifically. I have a short torso, and rental wetsuits and BCDs are made for someone taller than I am. I learned early that it's super-annoying to be tugging on a wetsuit or have a BCD up around my ears on a dive because they don't fit me. Also, having my own gear allows me to get familiar with where valves, clips, pockets and such are placed, so I don't have to waste time looking for something on a dive (or worse, in an emergency!).
Though I've had to redistribute things between my dive bag and carry-on more than once to meet an airline's weight limit, I've never worried that much about a bag fee, because the cost of renting gear at the destination over several days adds up fast and can total more than what an airline might charge for my bag. I'll gladly pay a bag fee to have my own gear with me!