Based on what you are showing me I don't see how that conclusion can be drawn and it looks kinda like a play on terms. As you have seen NH actually asks you go to them post denial for information and you can even go talk to them with lawyer in tow if need be (try that with the feds).
If an incorrect denial comes from/through the NH POC, maybe it's not contested via a process called an Appeal because they don't use that term, but it can be corrected via an Ammended Denial is all.
Permits and Licensing, Support Services - Division of State Police, NH Department of Safety
www.nh.gov
No the terms are clear, and if you bothered to look at the stats you may even understand.
Pay close attention to the " " marks as this carried specific meaning.
To them a "background" check is NICS and "research" is State. If you are denied by something in the NICS check, which Fed law requires them to run, you have to "appeal" through the Fed system.
Now if you pass a NICS check, under the Fed system that's it APPROVED. But under the State system they do additional "research". This is NOT required by Fed law, they just want to do it. If you are denied based on that additional research, which is not required by the Feds. Then yes, they can fix that at the State level, They don't call it an "appeal" because they define that as the Fed appeal process, so they call it an "amended denial".
Now if you look at the stats I posted for 2019 (the last full year at the time I request the info).
There were 53, 010 submissions.
There were 386 "Instant Denials", but don't be confused by the name. They defined this as denials based on NICS background checks, with no State "research". "Instant" is a reference to the "I" in NICS, that's why its in " ". Unlike the common meaning of instant which is immediate.
Then there is the "Denials" of 589, this is the TOTAL of "Instant Denials" and all other denials (a.k.a. denials based solely on State "research").
So with a little math we have 203 denials based solely on State "research". Those 203 PASSED the NICS check but were still denied by the State!
Of those 203, only 18 became "amended denials". That is, the State corrected their own mistake. And remember, those 18 had to prove to the State that the State made a mistake.
So 185 were wrongfully denied by the State! 185 that under the Fed system would have been approved.
There is one stat missing, the one the State does not track because its handled by the Feds, thats how many of the 386 denials were overturned on appeal. But since that number would be the same under both the POP and direct system of Fed checks, it's really not relevant to the discussion.
So you are supporting a system that;
Causes significant delays.
Cost the taxpayers $$$$$$$$$$.
Still follows the Fed/NICS process with the same restrictions and submission of information.
And wrongfully denies law abiding people their 2a right, 185 times in 2019!
And your only argument is that the Fed MIGHT change the rules. But if they do, that will all carry down through the state POP anyway.
Remember they work for us. 
