Latin didn't even evolve lowercase letterforms until the 9th century AD. "J" didn't evolve as its own letter until sometime in the 16th century AD. Early Roman had only 19 or 20 distinct (strictly uppercase) letters depending on exactly which scholar and century you're looking at.

IMO, the various letterforms in use by all of the alphabetic languages have more to do with the available writing technology of their day, and in keeping the letterforms sufficiently distinctive to quickly recognize whole words. Compare Etruscan engravings to early Roman engravings to German blackletter to Spencerian cursive to Zaner-Bloser cursive for example.

Hangul has a similar diversity in stroke shapes to Latin characters, but much more diversity in stroke position. I pity the poor engineer who might have been tasked with constructing an N-segment display for Hangul in the days before dot-matrix technology was feasible.

Finally, the 7-segment display doesn't actually look particularly recognizable when being used to render latin text. It only really works for the European evolutionary branch of Arabic/Hindu numerals. Even with only 26 (modern! English!) letters, we still have enough stroke diversity that we'd prefer to render text with more complexity. 2^7 segment combinations for only 2^3.3 characters doesn't sound so optimal any more, does it?

But if you want to stick with your information-theoretic observation, I'll point out the completely unrelated fact that some of the early and successful forward error correction schemes for Gaussian channels use rate 1/2 coding. Anything optimized for burst errors is quite different, though.