I'm pretty sure adding a rarity level to gun spawning in loot bins, would increase competitive-ness. Or maybe just keep this method in ranked play, but not "casual".
For an example off the top of my head:
SMG's and Shotty's uncommon.
AR's and LMG's rare.
Not saying that's how it should be that's just a quickly rushed example. A lot of other details and factors would play a role in deciding rarity and spawn rate. If a system like this is in place it's barely noticeable. From my personal experience every gun (not including care package) spawns pretty evenly.
All weapons in Apex Legends have a certain chance to spawn. So this kind off is already implemented.
P2020 and Mozambiques are more common than r99 for example.
The rates of weapon drops change per region, so for example, if you enter a named location you will have a higher chance on good weapons. Entering a region without a name is a low chance of good stuff.
You know in what kind of region you are by looking at the colour of the name at the top left corner when entering a new area. At least, it was like that on Kings Canyon.
You can find the different regions here (season 1): https://www.apexlegendsmap.net/?map=mp_rr_canyonlands_64k_x_64k
I'm not sure that they assigned probability values based on the type of guns as you have set them out, @SQ_GreenFire