Hi there, Casey! Firstly, yes you can choose to fold you're wheelchair instead of sitting in it - I did this for my first few trips on wheels. A few points to consider, though:

Even the lightest of folding chairs still has quite a bit of weight and bulk to it. If you are trying to get on one of the older buses, you will have to carry the chair up three (I think) steep steps, navigate it around a reasonably tight corner, then find somewhere to put it down without crushing anyone's toes. Also, you may have to do this with impatient people pushing from behind (yup, some people are morons

)

You will have to be resonsible for holding on to that chair, no matter what happens. If the bus has to stop suddenly, you need to be holding it tightly enough that it doesn't go flying into anyone. Though the brakes may help a little, they won't be enough to stop the chair 'escaping'. On a crowded bus, this task becomes even more 'fun', as you will have people pushing into, leaning on, or occasionally even sitting on, the folded wheelchair. I can't count the number of times I got whacked in the leg with my own chair, trapped a finger in the folding mechanism, or otherwise got injured because somebody shoved roughly into my chair.

You would have to do all of the above, no matter how tired or sore you were. You'd also risk having people banging into you, potentially flaring up any joint problems you may have (not sure of your condition, but I know this was one of the big things that made me change my mind).
And now for the alternative!

If you use the lift / ramp, you will have the space and time you need to get yourself on, with as little pain and energy as possible.

You won't have to carry your chair on, thus avoiding the risk of pulling a muscle, injuring yourself or others, or falling on the stairs.

If you stay in your chair on the bus you will be assured a seat, as you're bringing it yourself, so you can save your strength and keep you pain levels as low as possible.

You won't have to hang onto it, as it will be safely strapped down - no sore arms, shins, wrists or toes, and no risk of it getting away from you and hitting someone in an emegency stop.
I understand you not wanting to draw attention to yourself, or cause anyone any hassle, but it's actually better for everyone if you follow the reccomended Disney procedure. Loading and strapping down a manual chair doesn't take too long, especially if you're good at parking, and know where your safe tie-down points are. Even if you decide not to strap the chair down, please use the lift or ramp, so you don't risk hurting yourself.
As Goofy's Sis said, you may have to split your group up if the buses are busy, but you can arrange to meet in the Main Street bakery (or wherever suits you), and the first ones there can enjoy a piece of cake while they wait!
As for the people in your group being embarressed? I've got a few choice words I could say to that, but I'll keep things clean. As you have your own chair, I'm guessing that you have a long term disability. I'm also guessing that everyone you're travelling with, knows this. Why on Earth should they be embarressed that someone in their party is disabled, and is using the assistance provided by Disney, to make life easier and safer for everyone? Honey, I say you go and enjoy your holiday, using whatever you need to in order to get the most out of it, and never mind anyone who thinks you should limit your fun to save face. You'll probably find that hardly anyone notices you loading anyway - there are so many wheelchair and
ECV users at Disney, that everyone will be used to the boarding procedure after the first day!
Hope you have a magical trip!
