In my mind, yes, it'd be better to have random improvised weapons in the store from day 1 than the current "gotta research an axe to know what it does". Also they should be in the ufopaedia from day 1 too, of course.
XCom players research everything they get their hands on, because who know what the crazy modder decided to put the "axe" as a dependency for? So why risk it. For example:
There's a machete research project? I'm 99.9999999% sure my agents won't ever use a machete, but... I don't know if Solarius put the machete as a dependency for some kind of "uber machete to fight plant monsters" (along with say, a plant monster corpse?) research project, and I sure don't want to miss out on the potential "uber machete", so I research it. Of course, the immediate outcome is I get a ufopaedia article on machetes, they're crap, I roll my eyes at how silly it was to have to research it and I proceed to forget about machetes, safe in the knowledge that I researched it so if ever it was needed, it's done. Does it add anything to the game? no.. just some micro, the "Oooh research completed! Nice!", then "oh.. it's machetes, who cares?! I can't believe I had to waste research time on that..." which I'd say is actually detrimental to the experience as a whole (and probably why people comment on it).
I'd be very surprised if you ever had someone go "why do I know about axes at the beginning?! That makes no sense!".
About cluttering, that's a legitimate concern (and I remember complaints about that in Piratez). But the truth is, you've just delayed the cluttering but in a way that isn't making things much better, it's just cluttered later (once a player has researched all those cluttering items) instead of at the beginning since most player will research it all and researching doesn't feel good with the players. I can see three solutions for that:
1 - Remove the clutter from the mod altogether. But that's not great since it's good for enemies to have flavourful, and sometimes less powerful, items.
2 - Remove clutter by removing items that a player will never buy from the store. Have these items in the ufopaedia from the beginning so players know what the item does, and just add a mention that it is "unworthy of use by XCom" (but make sure that's true, unlike the hunting rifle that said it didn't measure up to other rifles even though when you get it, you have no other rifle).
3 - Gate the items behind some tech, so that the player isn't overwhelmed at the start. That can be "unorthodox fighting methods" as a day 1 research project and represents the agents pushing to get these items, or a "legal case for XCom excessive force in close quarters" which represents the staff fighting the bureaucracy so it accepts that agents will use axes/machetes/katanas in combat because we're fighting a threat that warrants it.
Given the bureaucracy angle you've got going already, I'd go for the legal case. Personally, I'd go for #2 but I know there will be players who want to fight zombies with axes and machetes, or want their Japanese agents to use katanas and then they'll go: "Why can't I buy machetes/axes/katanas?! Even a civilian can do that!!", So #3 seems like the better alternative. "Logistics" give you bigger teams in a van, "Personal protection" gives you kevlar vests and the "Unconventional Force" gives you weird items used as weapons. Now, not having immediate access to these makes sense within the story: You weren't allowed to by bureaucracy, which within the story world is established as heavy so it fits, but the staff pushed against the bureaucracy and now you can, much like every other project. Bonus: A starting player isn't overwhelmed by a million entries, they start with normal weapons and eventually get the unconventional one. They also don't get jarred by having to research modern terran tech.
