...There's bound to be s***loads of more advanced weapons available.
No, that's an assumption you're making based on a very possibly incorrect extrapolation of small arms developement to this day, into the future.
There's no way to really know how some certain branch of technology will advance, and at what speed - especially in a future like Firefly's universe, which most likely won't even happen.
There's actually two different assumptions: That there
are more advanced weapons - better enough to make a maningful difference between performance - and that they would be available. Neither of these assumptions can be proven correct. The first is a problem because it assumes that there's something still to be developed in the
basics of the small arms that makes it possible to achieve higher effectiveness - and I don't really seem how much more deadly small arms fire can ever be than it already is. It would require a lot stronger materials to construct stronger barrels, chambers and locks to achieve higher muzzle velocity by increasing the cartridge size and/or gunpowder amount and type.
Caseless rounds are probably going to be what peaks small arms developement for a long time, and their main advantage will be increased rate of fire and reduced price and weight of ammunition but as a disadvantage, they require a bit more complex weapons as far as I know. And the basic appearance of the weapons would stay much like it is now. Basic concept is same. Self-propelling rounds might be a next step, but again, more difficult construction -> less widespread distribution.
Tech developement is difficult to predict, but the way I see it, saying that small arms will advance to be significantly more effective from what they are now is like Roman gladiators saying that swords are going to be sooo much more better in 300 years. Sure, they designs changed but the basic premise stayed the same, it's a blade, how much more deadly can it be (in the hands of someone who knows their business). Sure, there were a lot of different stages in swords, different techniques (from cutting to stabbing to slashing to prying harnesses apart to light blades of the New era, and that's just the Western history of a sword in a nutshell) but the basic deadlyness of a sword was roughly the same ever since they moved from bronze to steel. Moving from bronze to steel in swords is kinda like moving from muzzle-loading firearms to cartridge based ammunition. Until something landbreaking comes along, I would predict that the basic effectiveness of small arms fire will stay roughly the same. You'd need to get something like vibroblade or light saber to make the premise of a sword more effective than it is - and something of similar magnnitude to improve small arms effectiveness in my opinion.
Mind you, there actually IS a weapon in Firefly that is ****loads more advanced than the conventional small arms used - in a way. But if I recall correctly, there was a reason why it never become widespread. I already mentioned about it in spoiler tags. It never managed to replace firearms, so there had to be some catch to it.
And talking about availability, the aforementioned weapon is one of it's kind as far as I remember. Might have been a few of them in existence, but the basic point is the same. Simple designs of modern day firearms are deadly and easy to manufacture compared to highly sophisticated weaponry, and no matter how much better the new designs were, the difference in lethality is going to be marginal at best as long as the
premise of small arms stays the same, so cheap, replaceable, durable and deadly weapons like shotguns (sawed-off or not), handguns, rifles and simple assault rifles like the AK-47 basic design will stay effective enough to be worth using. Both the Alliance and especially the Browncoats would have benefited from cheap and easy mass production more than sophisticated designs that were somewhat more effective. And of course, civilian weaponry will always be more variable than military weaponry...
Also... what Mobius said. Arrows even penetrate body armour better than assault rifle rounds. Are they more or less effective?

But I think this is well enough of this tangent...
