It is somewhat strange, but throughout all 3 films, he only grabs and keeps dropped weapons during a single battle, in the chase after Santonio in the Art Gallery (and even then drops them when no longer needed), and that is probably because he only had 7 bullets for his own weapon
Picking up a random weapon might be a bad idea for several reasons
- Taking up a few seconds that could be used to run further away, there might be 20 guys coming up that staircase, and a single pistol doesn't have that many bullets
- You don't know how many rounds are in the magazine, it may be an M4 and it may hold 30 rounds in the magazine, but what if the previous owner fired all but 2, you expect it to last and it doesn't, now you've got a stick... not bad, but not ideal in a gunfight, maybe going in knowing you've got nothing is better for John
- Maybe John's holsters are fitted for certain weapons and therefore they won't fit other pistols, this is reasonably common to be honest, so picking one up means carrying it in public
- Maybe John didn't like them, he has a certain manufacturer that he likes or trusts such as Glocks in JW2, and the bad guys are all running around with Berrettas.
Maybe its a little bit of all these reasons and more that the character of John doesn't pick up the bad guys weapons, I know that most if all of these reasons are slightly contradicted in various scenes through all three films but...
Out of universe
Having guns means there's no need for John to suddenly become a horse whisperer and have many requirements for horse kicking to death, or almost death in the next scene.
Having guns means no biker horse riding katana fight scene, and while ridiculous it was very well done (he does grab a gun eventually but before that).
Guns are a VERY powerful tool, so they tend to be ignored or nerfed severely in films to make the action seem more intense.
In the final battle, John needs armour piercing shotgun shells to kill the sweeper teams. Yet he was using an AR-15, a 5.56 rifle which has very good penetrative ability and he was shooting people in the head through clear goggles which would definitely NOT stop a 5.56 round.
So they made the armor "better" or "improved" to make a better-looking scene.