Suppression needs changed

 I had to purposefully miss my targets in order to attempt to suppress them,

This is why suppression is a fundamentally flawed mechanic in an arcade game. This may work in real life, but in-game there's no fear of death as you can just respawn almost instantly. It works way better in games where the death penalty is more significant. Battlefield is not that kind of game though.

There seems to be a minimum range to suppression, and maybe an internal cooldown on the application per target?

That seems quite plausible. You don't want suppression to kick in when you're in a CQB firefight after all.

There should be an actual impact on accuracy and ability to aim at a target when suppressed. 

Disagreed. It was terrible in Battlefield 3. It  didn't achieve what is was meant to achieve. So they chose to reduce it in Battlefield 4, but then it just became an annoying gimmick that added nothing to the game. So why try to force something that has proven itself to be a flawed mechanic?

The idea that you shoot back at a sniper that's hundreds of meters away from you is flawed to begin with. The logical action you take when somebody has a weapon that is better suited to the engagement distance is to disengage. Because even if you have suppression that affects weapon behavior, you still don't know whether you've successfully managed to suppress the sniper. In essence you're just doing something that is likely to get you shot in the head. It actively entices you to play in a way that gets you killed. 

Besides, why should you ever be rewarded for missing your shots? Let's say you consider snipers to be an issue, then why would you solve it by using suppression to begin with? There are many other solutions that would provide a much more believable experience, such as the ability to call in mortar/artillery on a certain location. An alternative solution is to flank and to close the engagement distance, to the point where your weapon has the advantage.

In Battlefield 6, suppression stops health regeneration. It's essentially telling you that the purpose of suppression isn't to counter snipers, but to prevent the enemy from being able to push ahead effectively.