@r1! Let me share my experience. I don't know if have told you this before, but when i got my bike (used) approx 5 years ago ... i had to change both of its front and rear tyres. Because:
1. The stock tyres were badly misaligned
2. The rear tyre was brand new but was slippery as hell. It was later when i found out the reason, it was because of its "rubber" which had gotten "rigid".
So your tyre may "Seem" fine by seeing its tread, but it may've lost its road grip because of that. If thats the case, then you will have to change the tyre ... to avoid any big accident (ALLAH na karay).
BUT FIRST .... do this. Reduce the tyre pressure in rear tyre to 28psi, and 20psi in front. (Do note that you won't be able to do pillion riding with this, but bear with me on this). Now take some test runs and let me know how the tyre reacts. If it still feels twitchy, or feels like its gonna slip any second ... then its definitely the tyre.
Do let me know what you find.