Happy Christmas to you.
I've done some investigation over the holidays and even disassembled the top part of shutter mechanism but decided to leave it alone in the end. What happens is that flash circuit has two cogs that when engaged are closing it (causing the flash to fire) and when one of them goes off, it breaks the circuit. When the shutter is cocked - one of them is engaging right away (the one controlled by second shutter curtain - I'll refer to it as C2) and the other one is off (the one controlled by first shutter curtain - I'll refer to it as C1). When the mechanism is fired, and the first shutter curtain is fully opened it triggers C1 to engage to close the circuit and then as second shutter curtain is released, it takes C2 off breaking the circuit. On shutter speeds 1/60 and faster the C1 never engages far enough to close the circuit. This is controlled by the mechanism that engages the shutter delays on slower speeds and it has differently profiled bumps that control how far the cog goes. Unfortunately on my Kiev-60 it is rather badly made and the cutout that engages on 1/30 is a bit shorter (than it should be as I can see from my Kiev-6C) so it almost always not engages far enough.
After some pondering and experimenting with shutter mechanism and having difficulty to put it all back, I decided to leave it alone. May be one day I get some other cheap Kiev-60 or 6C and get the part from there but I've given up for now.
Thanks for the help.