That are very hard conditions to improve skills. With saving during combat turned off my soldiers hardly survive long enough to gain some points this way, let alone a noticeable advantage.
Let's say one of them happens to survive 15 missions with 30 kills with an assault weapon. A very formidable performance, but he would gain mere 6 points on assault skill and 2 points on accuracy. While this would make not a great difference at all, most of my men die even earlier.
Depending on how new recruits look like (average stat?) I would increase the rate somehow like this:
A possibility to increase 1 point in the respective skill for every (!) mission with at least one kill or hit. The more hits&kills the better the chance. Maybe even two points if the skill is below 70. Basic chance should somehow be weighted like you did.
On a quick and very unbalanced approach the chance to improve skill by 1 could look like
if KILL>1: Percentage = 40 + 10*KILL + 5*HIT
if KILL=0: Percentage = 10*HIT
maybe multiplied with a specific factor depending on skillpoints and the skill itself.
I guess you are planning other ways of raising skills, but the main impact should come from combat experience. Medals should have some influence, too. I've read somewhere about a simulator, maybe the improvement like normal missions divided by 3.
I like the idea of implants. If they come early to play and improve with further research, improvement through mission could be lowered.
Anyway, why is skill improvement currently so slow? Or is it only me