I totally get how you feel. It's hard to let our little ones go, especially on a bus. The bus driver could have been a lot more polite, but who knows what they are taught to say/do. In their defense, I do have to say that they get in trouble for being late on a route, and if they are delayed by every parent of a preschooler or kindergartener, they'll never get to their destination on time. I do think the school should have provided you with information - for example, our schools assign older kids on the bus to help guide the littlest kids, who sit in the first few seats closest to the driver. But that didn't happen in your case, so you were understandably nervous about putting a little one alone.
That said, it sounds like you've done a great job raising an independent child - she's happy to go back to the dentist's office and so on, and I think you might consider letting her have some of her regular pediatrician appointments on her own too. Yes, you're there for questions and filling in the doctor, but kids need to learn to be examined by a doctor/dentist without parents there. My stepdaughter's daughter was horrible at the dentist at age 8 - I had to hold her hand, and she really wanted to me to recline in the chair and have her able to recline on my lap. Yikes! So I think dentists may try to prevent that by getting the younger ones comfortable with solo exams when there's nothing much going on (just some gentle probing, some suction and squirting of water, and some cleaning). Then when they need more (like a filling), the kid is already a pro at the chair.
One time my son had to fly alone from Florida to Massachusetts - my son and I were with my mother in Florida, and my husband was at home. My mother's brother died, and I had to send my son home without me so I could stay. He was about 7 and I started to lose it when he marched his little independent butt down the jetway, chatting away with the flight attendant escort. He was fine, I wasn't. Normally I'm pretty composed, but this was a lot. I wasn't worried about the plane crashing - just about him being nervous and about my husband not being able to connect with him. My mother (who is very critical 90% of the time) was very supportive and told me he was doing fine because I am such a good mom. So that helped a lot. That's why I'm telling you that you should be easier on yourself - if you have an independent and confident child already, you are way ahead of the game! This will serve her very well in the long run.
So I'd try to let this go. The driver has no control over the regulations (probably insurance stuff, and of course there have been news stories about parents getting on the bus to solve bullying problems themselves, and it never ends well). So the driver wasn't attacking you personally, just following the rules, and while he may lack a lot in personal skills for public relations, that doesn't mean he's a bad driver. If he makes the kids follow the rules too, staying in their seats and not throwing stuff and not swearing, that's what you want.
So your feelings are completely normal, but you will get through this. If your child is having fun and has mastered the skills necessary to get herself to school and back, that's great. If she has a tough day now and then, that's normal. That sometimes hits on the first day, and it sometimes hits after a month when the novelty has worn off and they figure out that they have to do this stuff every day! Welcome her home, don't tell her how much you missed her, but do say you hope she had a good day. If she can't tell you what she did all day ("We had snack" might be all you get), please know that this is completely normal and what you are likely to experience for the next 10 years! You'll learn more at school open houses and teacher conferences. Do go through her backpack and see what things she brings back, and you'll get an idea of activities.