The avatar appears in two places. Pardon my shameless plug, but what about a second choice of generative avatar/glyph that replaces the square box with something like this:
Treat this as a very crude idea. The ENS name appears in a unique place depending on certain features that one can define, the colour can be used as a trait etc. One can take it a step further and allow a mintable high-res version of this glyph with scrambled characters replaced by those of user’s choice + a generative GIF capturing the process (see low-res sample below; high-res version here). This could be done at a fee and proceeds for this can be routed back to DAO treasury. All of this could be made a choice of course and not enforced. People can keep the default version or choose a new basic glyph for free if they like or mint a set of high-res glyph + generative GIF for some price.