Agreed: I had a few runs to refresh my memory and the game is quite generous with weapons and with time, too. So, I suspect that you are indeed right in the "no-bullets" challenge being a "covert" intention among the designers' intentions. I will elaborate, with the risk that I am simply writing down what you do in the video
(tangent: actually I could check from work as we have VPN, but I will be doing that on Monday: "scholarly research purposes", indeed!
).
Stage 1 can be easily cleared with a "no-bullets" approach and I even learnt that, back in the day. Stage 2 is tricky if the alternate sections involve a hostage situation: saving the hostage girl only with punches requires a lot of patience. The gallery shooter sections are simple: just dodge bullets. The boss is tricky but doable, indeed.
Stage 3 involves tricky jumps and risky situations because there are a lot of environmental obstacles, and the boss requires a specific technique (well, just dodge his shots if you cannot punch him quickly, right?). Stage 4 has a lot of sniping enemies and peculiar situations (e.g. the enemies shooting from above), so time should become an issue. At 120 seconds per stage and when pursuing a "no-bullet" challenge, stage 4 should become quite a challenge and leave very little room for mistake.
Stage 5 can be cleared in the default 120 seconds and with one weapon refill, but players really need to advance steadily. If the RNG is unfavourable, players will not get an extra 20 seconds (i.e. the "clock" power-up) until the very end of the stage. With a "no-bullets" approach, the stage becomes a hard challenge if only because every punch/roll attack must succeed and players need the clock bonus to be on the safe side. Stage 6 sums up all those challenges and adds the final boss battle, which indeed revolves around throwing grenades left from zakos.
On top of all this, the bonus sachet of...something (ah-hem!) should be worth 300 points if collected immediately, but worth less (100 points, then 50 points) otherwise. RIght now, I am also getting the impression that they appear more often when a player uses physical attacks. So, a "no-bullet" run would offer more scoring opportunities, force players to be very aware of the clock, and engage with stages' layout and enemies' positioning in a much more precise manner.
If all of these ruminations are correct, you should indeed be right on the money. That would be great in my view, as it entails that the game has something of a "deeper" design than I always thought. A loose connection is that
Thunder Fox should have been produced by roughly the same Taito team and emphasises the "no bullets" approach and speed runs in a more explicit, score-driven manner (i.e. time bonuses and weapon/bullet bonuses are considerable). Ironically, it is a more "pure action" style of game, at least on the surface.
Crime City is still not as strict as the two RT games, of course, but that is a good thing in my book.
and re:
sly spy (ah! I am chuckling as I write!), I love DECO but their idea of this micro-genre was, uh, more Rambo than Bond, given how much ammo, special weapons, etc. the player can get
"The only desire the Culture could not satisfy from within itself was one common to both the descendants of its original human stock and the machines [...]: the urge not to feel useless."
I.M. Banks, "Consider Phlebas" (1988: 43).
A serendipitous arcade trilogy reunified. Vs System port a reliable USP, as always; FC original is on M2's beautiful Namco Museum Archives Vol.2, so that version's well-catered for, even if you don't have the cart!
Never caught that. Seems, uh, poignantly instructive... top ten Nasty Words not to call a bloke armed with a knife? 
