The rules do allow such a readied Sneak Attack

The wording of that passage in grease is, in my mind, very unclear, but readied-action attacks and attacks of opportunity are at least two cases that clearly do work. They take place during the movement, so they definitely work. Readied actions basically take place whenever the person who readied the action wants it to, so you can definitely ready an action for “as he moves past me” rather than “right before he starts to move” if you like, and clearly the Rogue would like to.

As you note, this is a really difficult maneuver to use effectively. Worse, as the Rogue levels, there’s very little he can do to improve it; stuff like Two-Weapon Fighting et al., iteratives from extra BAB, and the like, aren’t going to work with it. So as he levels up, instead of getting to improve this tactic with his feats and other choices, all he can hope for is extra Sneak Attack dice – which aren’t going to keep up on their own.

Note, though: “he moves and remain at 5ft of the enemy in the grease area, and ready a 5ft step + attack with the same trigger,” doesn’t quite work: he could only ready that action if he had not moved already in the round. If he was already in position and did not need to move into it, then it would work.

I’d strongly consider reverting to the 3.5 rule

While grease was and still is one of the best possible standard actions you can take, this particular facet of the spell was a good thing, in my opinion. It encourages teamwork and is an excellent example of a Wizard helping his teammates shine, rather than stealing all the spotlight himself. If I was to nerf it, rather than removing this aspect, I’d most likely tone down the spell’s double-jeopardy nature, or its effectiveness even when the enemy saves and makes his check.

This particular rule is thus more of a nerf to Rogues (who, after all, could have easily gotten a wand of grease since as a 1st-level spell it’s fairly cheap, and they have UMD in-class) than it is to Wizards (who still have plenty of options). Rogues don’t need a nerf. In general, Pathfinder makes it decidedly more difficult for Rogues to get multiple Sneak Attacks per turn, and I don’t think that improves the game at all. It was, in fact, already too hard for them to do this in 3.5, in my opinion. So I’d strongly consider reverting to the 3.5 rule.

The answer is no, the creature is not provoking an attack of opportunity--Seize the Moment is merely providing an AoO for something you normally wouldn't get. Contrast this with the wording of Outflank (taken from pfsrd)

Outflank (Combat, Teamwork) You look for every edge when flanking an enemy.

Prerequisite: Base attack bonus +4.

Benefit: Whenever you and an ally who also has this feat are flanking the same creature, your flanking bonus on attack rolls increases to +4. In addition, whenever you score a critical hit against the flanked creature, it provokes an attack of opportunity from your ally.

The wording of these two feats is very different, and you can see via the italicized portion that if seize the moment were intended to work the way your friend describes it would also have to be worded this way.