What would make you think it is a proof? I don't believe there were 1965 proof coins. Yours is a normal, but very worn, 1965 quarter.
First, welcome to the neighborhood! @l.cutler is correct. Proof sets were not made in the years 1965-1967. Instead, the Mint produced Special Mint Sets.
“There is a wide separation between the “E” and “S” of “States” on Proof Reverses but the letters are nearly touching on business strikes. The lettering is bold, has flat tables with squared corners and edges compared to regular issues. The leaves are well-defined and the leaf opposing the “A” in “Dollar” nearly touches it, while a separation is present on regular reverses. Also, the leaf before the arrow bundle upon which the Eagle is perched curves slightly to the left. It then rises above and obscures the arrowhead points on the Proof die version. On business issues the leaf terminates noticeably lower and leaves a gap between itself and the exposed, uppermost arrow point” https://coinweek.com/modern-coins/us-coins-proof-reverse-washington-quarter-pt-2/
In 1965 there were 1,819,717,540 business strikes and 2,300,000 Special Mint Sets struck https://en.numista.com/catalogue/pieces55.html
Ok, your article references earlier business strike silver quarters that were struck with a reverse proof die. As stated no proof coins were minted in 1965. A quick google search brought up that a new reverse design was used in 1965, yours is normal for the year using this new reverse.
Welcome to CT @Kean. As already stated, there are no 1965 proofs, so it couldn't be one, but I like your analysis and detailed observations as you listed them in post #9. Keep looking and good luck.
What you are describing here is commonly referred to as the "Type B" Reverse (or RDV-002) which was a proof die reverse that was used on some Philadelphia business strikes from 1956-64. There was no RDV-002 coins in 1965. The 1965 did actually have three different reverse varieties (RDV-003, 004, 005)...but all were different than the silver proof reverse.
The coin has had its rims removed with a knife. It could be a file but the unevenness implies a blade.