"Do you want the cops to follow every single person who has purchased multiple firearms in a given period?"

I'd defer to the authorities   to determine how best to use the data. Maybe the behavior would trigger an additional delay or an interview. Maybe certain behaviors are more indicative than others. No matter what, more data could only make them more effective in their work.

As to your latter point: criminals get guns on the black market because straw purchasers easily elude police. Tracking sales stems the tide of illegal guns because of this, so over time it would be much harder to illegally obtain a weapon. Right now, it's ridiculously easy and it's all because we don't give enforcement the tools they need to put bad actors in jail.

Anyone obeying the law would suffer a minor inconvenience and/or expense at worst. How many lives is that inconvenience worth? There's no question that more data would assist with prevention, though. As I said, that part is common sense.