Yes, cheaper products often will work just fine. I, myself, have used Tide a lot the last year--but only because A) I prefer unscented, and it works better than unscented versions of All or Arm & Hammer, even for my typical load that merely needs freshening, and B) reasonably good sales at one store.
But if I were willing to use scented detergents again, I'd have more choices--and more than likely, Tide would be more than I'd need. Sale price or no sale price at that one store.
Indeed, I've used cheap dish detergent a lot. Yes, the more expensive stuff probably cleans better and lasts longer. But for my usage, cheap is good--I typically only wash a few dishes at a time. I don't need to worry about 2 drops lasting for six mountains of dishes from hell.