It’s a little different for guns, they’re more tightly controlled and there’s often paper trail of their purchase and licensing of owners, so your example doesn’t apply.
I think there’s also the issue that you’re more likely to be murdered by someone you know than a random person. At the very least, matching bullet marks from shots fired from his associates guns to any casings found at the scene is just due diligence.
The reality of guns in the US is that a gun linked to a crime is likely to be part of a cohort of guns with a "short life to crime" having bounced through obfuscating straw purchases.
The most rapidly increasing (although still small in absolute numbers) class of gun associated with crime today are the 3D-printed variations:
Thousands of guns are found at crime scenes. What do they tell us?
So, yes, if this a crime of passion, a dispute between aquaintainces that escalated badly then there's a good chance the gun used has a history of ownership and registration.
If this is a crime related to home invasion gone badly then it's more likely to be a gun that fell off the radar some time past.
You'd also be excluding everybody who illegally has a firearm or knife or whatever the murder weapon is.