Oh man, I feel for you on this one. I'm definitely the type to rag on myself and my riding lessons, so I can feel you on the negative attitude.
Honestly, it sounds like you're craving some independence, and that's something a lease would definitely help fix. It also sounds like you're doing exactly what I do when I've been in "all work, no play" mode for too long: your focusing too much on the progression of your riding skills and not enough on your love of horses. I'm very goal-oriented, so I can sometimes lose sight of the journey because I'm staring too hard at the goal, you know? You feel pressured to use what little time you have to learn as much as possible, so you push yourself and then get upset when things don't progress as fast as you expect. The only solution I've found is a lot of self-reflection and a return to simpler things (trail rides, light hacks, just enjoying my horse's company, etc.).
If I'm really having trouble, I might take a few days off (so a couple of weeks, if you ride weekly).
It's great that you've asked to ride one horse for the rest of the year. Having a horse that you ride consistently can really improve your riding (not that riding different horses doesn't, just different types of improvements) and can help you feel more connected to that passion for horses that lurks inside of you. I'd also keep your eyes open for a lease (especially one at your current barn, so you can continue to take lessons there). Maybe even ask your instructor about a possible lease? I know at my barn you can choose to lease one of the lesson horses and then you schedule times to come out and ride on your own two or three times a week, on top of your lessons.
In terms of the money argument for not leasing... well, I'm probably the worst person to ask about that, lol. I'm always of the opinion that if I have to choose between saving money and doing something that makes me happy, I'd much rather be happy than rich.
As for your connection with the other, younger students—pssshaw! I'm 21 and lesson at a barn with students that are mostly in the 12-16 year old range and I have no troubles relating to them. After all, you've all got one major thing in love: horses! Try asking some of them how their lessons have gone/what their show plans are/what they think about different disciplines, or tell a funny story about something your horse did in your lesson. Instant conversation.
Time wise, things will get way easier when you have your license.
Side note: I like your idea of having a "fun" lesson. Maybe talk to your trainer about organizing a Pony Club trail ride or something like that, too! One fun ride together can make all the difference, especially if you guys feel like you're stagnating in the learning department.