4 options I think could work for this situation.
1. we leave it as it is, but increase spawn rate of guns at drop a smigin. 1 gun should be in at least every 2 buildings.
2. Have everyone drop down with a p2020. It's the simplest pistol, not a lot of people use it, and it's not that strong of a gun. Meaning people will be aiming to replace it quickly. So you will still see the scramble to find guns, but it wont be quite as dire.
3.Give everyone a new, only given on drop pistol. It uses no ammo that can be found, and it has weaker overall stats than other pistols. Would still let people go to town on each other at start, but almost everyone would still rather be scrambling for a better gun, then try and get a kill with the Psh02 - 'pea shooter'
4. Let each legend spawn with different pistols. Each legend would be given a specific starting pistol, like the p2020 for Lifeline, or the Mozambique for Gibraltar. Key there would be don't let anyone spawn with a wingman (unless it gets nerfed.)