Well, that one doesn't work for me for some reason(I keep using normal attacks with that), only the jump variant does the trick.
I had Blade, block, acrobatics, athletics and hand to hand pretty much above 100, still nothing. Kicks still won't happen againts enemies in my crosshair

So in your case, it actually works? :mellow:
Hey there, Smert. Your problem is very familiar to me, so I figured I'd chime in (disrupting my lurking habit). A few weeks ago, I started running Open Cities Reborn, the newest version of which requires OBSE v20. I'd been running Deadly Reflex 6 for a couple months already, always using the stable OBSE v19, but I figured there wouldn't be any trouble if I used OBSE v20 Beta 5. That's when the problems started.
I loaded up a new character and started playing, but I noticed a couple oddities. Every once in a while, my shield bash wouldn't activate. My headbutt animation was broken, and my character would swing her sword, but it sounded like a headbutt when it hit my enemy. After a few hours of gameplay (and after leveling up my character via the console so I'd have access to kicks), I started noticing the same exact problem that you have. If my crosshairs weren't looking at anyone, then I could use Block+Jump in order to perform all or most of the acrobatic moves, but I could not use any offensive kicks against an enemy. I'm pretty sure that I could also use Block+Attack+Back in order to do a backward dodge. My Bash didn't work anywhere, unless I loaded a much earlier character (straight out of the sewers), same with weapon throw and the stationary dodge.
It took me a while to figure out what could even be related to the problem since I'd only loaded up some quest mods and Open Cities Reborn. Eventually, I decided to try uninstalling OCR and changing OBSE back to v19 and BAM! Problem solved, DR6 went right back to normal. I'd been looking through all the DR6, OBSE, and Nifse forums in order to figure out if my problem had ever been encountered before since I've really been wanting to play the latest OCR, but I'm too afraid to mess with my current setup. I know that it has to do with the difference between OBSE v19 and v20 Beta 5 since I encountered the problem without OCR and with OBSE 20, then it was fixed when I only changed my OBSE version. I personally run DR6 with 1st Person Legs, Nifse, Drop Lit Torches. I don't use the HUD addon or the blood spatter. The only changes to my DR ini are changing Magic Crits to 1% of the original value.
So TL;DR are you using OBSE v20? If so, see if you can manage to change it to v19 (other mods permitting), and then see if that works. If it does, then maybe we've discovered the source of the problem.