Lynx is FATE's avatar, but he doesn't act like a supercomputer at all. He's nasty.
And Lynx should have killed Kid and taken the Time Egg to control time. Why didn't he do that? It's not like him to keep her alive.
Big plot hole.
Actually, most supercomputers in the history of fiction have been nasty. Think of the, whatever he's called, in TRON. Not exactly the nicest of programs. Take the actions of the programs in the Matrix, also - whether the Architect, or the rogue Agent Smith. Both are diabolical in their own fashions. Lynx is an avatar, but that doesn't mean he must act like a computer. The wishes and basic ideas and all of the computer were put into him, it's true, but he still functions using a biological mind, which is likely more complex than even the advanced FATE computer. As such, Lynx acts more like a sentient biological being than a supercomputer.
And for the last... the Time Egg cannot control time. The Time Egg is, if you remember Trigger, the potential to do something. That something isn't always what the user wishes, either. It doesn't work like a master-key to time - if anything, the Flame has of that sort of power - but rather like, well, a trigger. And a specific trigger at that. The first one had the power and potential to save Crono. Since his allies put enough devotion into its use, that potential was realised. However, if they'd gotten it into their heads to use it to change what happened with Lavos... well, it couldn't have done that, I don't think. Likewise the second one had the power to go to the Tesseract, but nothing else. They are not a technological tool, after all. They are magical, and I think have a certain will of their own. Like the Masamune, in a way. The Masamune might have the power to do great things, but only certain great things, and only if used in certain ways. It is not an unlimited tool. In Lynx's hands the Time Egg probably wouldn't have even worked. The crafts of the Three were rather strange, after all - more like magic, less like science (though to them it all probably seemed rational.) Melchior's famous Red Knife; Gaspar's Time Eggs. They're magical things, and about as predictable as Sauron's One Ring. Of them all, Belthesar was the most scientific (as we know science), what with his Blackbird, his Epoch, and his Chronopolis.
Finally, it may just be a trick of memory, but... I don't think Schala had a true Time Egg. Her Astral Amulet contained Lucca's incomplete attempt at one. It, too, was linked to certain things, and certain things only (ie. the protection of Schala.) The Time Egg that grants access to the Tesseract... was that not given by Belthesar atop Terra Tower? And by that time, FATE had been long destroyed.
