I think it depends on the relaxer. Some rel hair relaxers will eat through my hair without straightening it while others don't straighten enough. I think it depends on what works on your hair. For me, personally, I had to judge each relaxer I used on its own merit, regardless of whether it was lye or not. I find that most "boxed" relaxers are terrible for me.
In order for you to get an idea of what I mean, I've included some personal reviews of relaxers I have used in the past.
Motions Lye- this relaxer was terrible. I left it on to process for 45 minutes and it did nothing for it.
Revlon Realistic Lye-this relaxer was also terrible. It had the OPPOSITE effect. It ate through my hair within 5 minutes.
Revlon No Lye- wasn't so good, either. Left my hair brittle and damaged.
African Royale No Lye- this was discontinued. Straightened okay, but results were not long lasting.
TCB no lye- did nothing for my hair. Acted more like a conditioner than a relaxer.
Creme of Nature- straightened well, but not long-lasting. When they changed the formula (from lithium hydroxide to calcium hydroxide) it's performance wasn't as good.
Dark and Lovely- didn't like this stuff at all, even when they changed their formula. Results didn't last and broke my hair off.
Bantu- I'm still psychologically traumatized from the burn marks I got from using this stuff.
Just for Me Kids Relaxer- Very good product. Left my hair straight for about 3 months before I needed a retouch. Highly recommended for people on a budget.
Phytospecific, no lye but does not contain any of the "lye cousins" (calcium hydroxide, lithium hydroxide or potassium hydroxide)- I'm saving the best for last here! Yes, it is expensive, however, because the results are long lasting, I don't have to relax my hair but twice a year--once a year, really. I had my hair relaxed in December 2005 and about two days ago, I got my hair washed and set and it still looked like freshly relaxed hair. Even the roots were manageable. My hair contains more movement than it ever had in the past and it's never frizzy--even in the summer time.