Well, actually it sounds like some of the anxiety might be due to your knee. I know when I have had anything going on with a leg or foot I feel insecure getting up on a horse.
I was instructed as a kid to not get on how most people get on, and to dismount differently as well. We were taught to take up the reins with the left hand and adjust the rein length so there is a 'feel' of the horse's mouth. We were also taught to turn the horse's head to the right, first to prevent that oh-so-friendly nip on the backside, but also so if it tried to take off while mounting we could very easily keep it moving on a small circle.
We were also taught never to be too proud to ask someone familiar with the horse to hold it while we got on(as long as they were sklled and not about to make the situation worse). If we didn't know the horse and no one was there that could hold him, we were taught to face him toward a corner to lessen the chance he'd take off.
I saw one very athletic lady 'cheek' a horse and hop on. I was young and impressionable and never forgot it, but I doubt I could do it these days. She took ahold of the bridle cheek with the left hand and swung up in to the saddle while the horse was running around in a small circle, LOL!
We were always taught to get on VERY quickly, as well. We were also taught by a lady at one point, who INSISTED we get on with a mounting block. She said that mounting and dismounting puts an awful lot of wear and tear on the saddle and girth and she absolutely insisted we not get on from the ground, even with the large ponies where not all kids needed a mounting block. Her other argument was that horses don't have a chance to get restless or stir up trouble when one mounts with a mounting block.
I still get up very quickly though. I don't flop down into the saddle though - I was also taught that tends to create horses that take off when the rider mounts.
The way we were taught to get off that was different, was to take both feel completely out of and away from, the stirrups before swinging off. I've seen quite a few people dragged when the horse startled while they were dismounting 'step down style', and I think not being free of the stirrups is a very bad idea when dismounting.
As far as anxiety though, we were also taught that we should feel from the start if we can handle this horse - trouble restraining the horse to mount was one indication that we should not get on the horse. As a young person I was like all teens and young adults who feel they have to get on everything. I no longer feel that way.
How's this for getting on quick? This guy is an old fashioned colt breaker.
And some of his disciples:
Or here's another way to get on: