I like all that stuff. Going up against crazy government conspiracies, old guys playing at being the Illuminati (badly), and finally assaulting a whole skyscraper packed with corporate goons is cool, larger than life action movie stuff. Yes, it has a juvenile pulpy vibe to it, but that just keeps things shaken up. It got unbalanced with the Colt Commando and the rocket launcher, but then having your hands on heavy firepower also felt very satisfying in a way the second game's more fine-tuned weapon arsenal never did. If there is a weak part to the game, it's fighting through a million tenements teeming with your shotgun - fine on its own, but dragged out too much.
The second game's main story-related problem was that it never went anywhere. The "film noir love story" tagline mainly meant Max was left wallowing in his depression and misery, never doing anything about it. The main opponents being a Russian gangster and a group of people working under the guise of a cleaner company doesn't pose an interesting or active opposition, it is not really inspiring. Meanwhile, Mona Sax was a case of taking an interesting side character and removing the individuality that made her interesting by spinning her into Lara Croft / generic action chick. While the first game had interesting subplot, the second was too focused, trying too hard to be that film noir love story. Put these things together, and the chemistry and rage that held MP1 together is missing. The game has too little momentum, and as mentioned before, the levels aren't good enough to carry it.
In many ways, MP was a victim of its own success - the first part turned into a hot IP, and invited the kind of tragic meddling that turns good game franchises into bad movie imitations.
[edit] Inspired by this thread, I've dug out my DVD today and replayed the first part of MP1. Still very enjoyable, and the level design has aged better than I suspected.
