I can't truly argue against any of the classic silver 50 type coins .... each of them in their own way represents a wonderful treasure trove of history and design .... often filled with stories of greed and political maneuvering.
Let's discuss the Albany Half Dollar ...
Minted in 1936 to commemorate of the granting of Albany, NY city charter. Albany is (was?) the 2nd oldest city in the country .... but this was still an event of local interest and not national significance.
Total mintage including assay coins was 25,013 ... eventually 7,342 were returned to the mint for melting ... leaving a total net mintage of 17,671 examples. Compare that total to the 1909-S VDB and let's discuss true rarity.

This has always been a collector favorite due to the excellent design work of Ms. Gertrude Lathrop. She followed on the next year with the design of the New Rochell Half Dollar.
The obverse of the coin features a beaver eating a maple branch. C'mon ... how many other USA coins can you point to that feature a beaver on the obverse? Historical records suggest that Ms. Lathrop actually kept a live beaver in her studio, courtesy of the NY State Conservation Dept, while working on the coin.
The reverse of the coin contains what I think is one of the best 'quiz' questions in
US coinage history .... "Which USA coin has the smallest size LIBERTY?"
Look real close above the three figures on the reverse ... there is an eagle flying above them. Look extra real close above the eagle ... and below the CH of "CHARTERED" .... there is the minuscule LIBERTY that Ms. Lathrop was required by law to place on the coin.

The problem with this coin from the OP perspective is that he/she will not find any examples in their price range. Current Numismedia price at MS61 is $294 ... and I have never seen a circulated non-problem example for sale.
Here is a photo of my PCGS MS65 Albany Half Dollar ... great coin indeed!

David
Edited for spllenging