At the risk of pointing out the obvious and maybe being whooshed, because the number would be meaningless. Saying Smith’s ERA is .391 will make people wonder what the hell that is.
The reason you calculate ERA based on nine innings isn’t because pitchers should always pitch nine innings, it’s because a baseball game is nine innings long, and so it compares the pitcher’s performance with how many runs a team typically scores in a game, offering a comparison between the pitcher’s performance and the normal rate at which runs occur. Since most fans just know a team scores about 4 to 4.5 runs a game, they know an ERA of 4.00 is acceptable, 2.80 is really good, and 6.10 is horrible.
I don’t think fans really know the average number of runs scored in a game. They know an ERA of 4.00 is acceptable because pitchers with that ERA are described as decent, and pitchers with 2.80 ERAs are described as great, and pitchers with 6.10 ERAs “stink”. Fans don’t know the average number of hits per game, but they know a .285 average is OK .330 is really good and .225 sucks.
If you changed ERA to a per inning value, it would take about a half a season for fans to get used to the new metric, and know that .445 is OK, .310 is really good, and .675 sucks.
Nothing, really, I’m happy with ERA as-is, I just think that a change to per inning ERA would be perfectly understandable after a modest transition period.