I have this string butler mounted to my Ibanez AS73T, with a Bigsby style tremolo. It has definitely improved the tuning stability of my guitar. This is extra noticable when I'm putting new strings on. I've had this guitar for 11 years, and have always had to spend a few minutes bending the new strings to get them to stay in tune. With the string butler, I'd say the break-in time has been cut in half. I still can't use the tremolo arm without putting the guitar out of tune, but that's probably because of issues in other places. I believe the remaining instability issues on my guitar would require some locking tuners and a new rollerbridge. So the string butler is an improvement for me, but not a complete fix.
It should be noted that the string butler only straightens out the 4 strings in the middle of your headstock. The headstock on my Ibanez is pretty wide, and both of my E-strings are still angled at the nut. I was aware of this when I bought it, but I still think it's worth mentioning.
Conclusion: If string slipping in the nut is the main cause of your tuning instability issues, I think this will help a lot. Especially if the outer strings are already going straight across the nut of your guitar. Just be aware that there are other factors that affect tuning stability.