Something valuable I’d like to add, not sure if is been covered already.
The British weren’t looking at 6” armed cruisers because they really wanted them, they were looking into them because it was the bare minimum they surmised would be able to do the job of a cruiser somewhat effectively. They were a complete compromise because it was the only possible way Britain could conceivably come out of the LNT with more cruisers than any other large signatory power, aka the US and Japan.
They tried (successfully) to convince Japan and America on the fact that while Britain would get more cruisers, a higher percentage of those ships would be less capable and lower tonnage 6” armed warships while both other nations would get less cruisers but a larger percentage of those ships would be the more useful 8” armed ships, therefore it would be fair.
Any serious planning before the LNT for 6” cruisers was largely based on the optimistic assumption that they would be able to convince everybody else to be onboard with a CL/CA tonnage split in such a way in the upcoming negotiations.
Something odd is mentioned here: 8 inch gun cruisers being called "Useful", which they were certainly not. The 8 inch gun was simply a political choice, rather than a practical one. The gun was too big to allow rapid fire and therefor was similar to the even bigger clumsy and slow rate of fire battleship guns, unable to effectively engage targets at short ranges, which was the cruiser's domain. The cruiser's weapon of choice (Until treaties came into play) was the QF gun type of calibers up to 6 inch, or around. This made these ships multi purpose in their nature, capable of doing a variety of jobs, unlike the 8 inch cruiser, which basically was limited in a role, due to its lack of capabilities. Most navies, except the Royal Navy, considered the 8 inch cruiser a substitute for the battleship, which was not allowed to be constructed due to treaty demands.
The Royal Navy in the interbellum demanded a large number of cruisers, which was restricted to allowed tonnage, so the British were not at all happy with the big maximum sized heavy 8 inch cruiser, which took up too much of the tonnage allowed to build. What was wanted was a multi purpose vessel capable of operating on its own oversea and in a trade protection role, something the Heavy cruiser could hardly perform in, due to its larger size and slower rate of fire guns. In reality the County's were deployed as such, but did nothing more a smaller cruiser could do as well, so the point is the allowed tonnage was not used wisely.