I mean... Final Fantasy VI had a few sequences like that: climbing the magitek armor to chase Kefka, fighting in free-flight during the ascent to the floating island, the second airship raising from underneath the waves; any of those would have been equally astounding... if FFVI had the same graphic as FFVII. The SNES just wasn't capable of giving scenes like those the kind of impact they deserved, but in terms of such things happening, FFVII isn't really that far off from its predecessor; that kind of absurd stunts are part of Final Fantasy DNA. FFVII just does it with better resources than ever before.
Although I do agree that the bike-riding minigame is pretty fun - and it has that much more impact because it's something NEW, in addition to being the first genuinely good minigame FFVII managed to offer. Not only is it the first time that FFVII approach to minigames of "create as many as possible and then double that twice, that way we're bound to get some good ones out of it" produced a positive result, it's also the first time within the game itself that the 3D nature of it is fully taken advantage of.
That said, I do think that, as the first instance of an "action-based-combat" situation in the Final Fantasy series, it's the harbinger of dark things to come... and I might add that the sequence is only good because it happens just once and strikes with the maximum impact. If it had been, let's say, the third time it had happened? That would have made it lose its shine, which I think would have greatly compromised the sequence's impact.