There is good riding, in which the rider can signal alot to the horse by seat and legs, and doesn’t necessarily always use the reins first to turn or to halt or to slow (but certainly does use them if the horse blows through the other aids). This is true in jumpers, dressage, and western performance disciplines. For instance, jumpers at speed need to turn off the outside aids, not get pulled around by the inside rein, or they will get off balance. You can half halt a well schooled dressage or jumper horse by sitting up and engaging your core. You can make a reining horse slide with weight and no reins.
This is just good riding. Unfortunately, it takes a while to be a good enough rider that your seat and weight cues can be meaningful to the horse. If you are wobbling all over the place that’s a lot of white noise for the horse to process, and a lesson horse that interpreted every bobble of a beginning rider as a cue to halt, to turn, or to canter would be too fine turned for the job. Beginner riders need blunt cues, and the opening or leading rein is the basic cue for turning, just like you really do put pressure on the reins to halt. Beginner riders need to learn these basic aids on a quiet and somewhat deadhead horse. When the rider has an independent seat then they can start to learn how to turn, slow, and halt a horse off weight and seat and leg aids. But the rein aids remain as backup, and as Dr. Andrew McClean points out, the rein aids are where the horse learns to give to pressure (negative reinforcement) and the weight aids are secondary learned after the rein aids. This is true when you break a colt. BTW I really recommend Dr. McClean for the best current overview of how horses learn and the different training modailities. Getting those straight in your head will prevent a lot of confusion later.
OP, as a beginner rider you need to recognize that you may be taught some things at the start that you will discard or modify as your riding improves over time. But until you yourself have a truly independent seat and independent use of the aids, you will indeed be more reliant on rein aids than a very experienced rider on a highly tuned horse.
I have found that horses just naturally learn the weight aids as you go along riding them, if (big if) your seat and weight are consistent such that you are always looking in the direction you want to go, and never behind or infront of the movement. I was amazed the first time I dropped my reins to the buckle and realized I could ride patterns with my mare just by looking where I wanted to go, because the weight of my head was enough of a signal. Now this only works if she is totally co-operative that day. if she wants to go beg carrots from someone standing at the gate, or wants to go sniff n squeal one of the other horses, then yup the reins come back into play big time.
but really I was just able to get more out of them, most likely.