ERA is reported as the total innings pitched, with the fractional part reported as a 0, 1, or 2 after a decimal point. Who decided to use this as a stat? When was it first used? Was there any debate about this method? In other words, who discussed it and what was the discussion?
Could you clarify your question, please? Because I think you are mistaken unless I'm misunderstanding what you mean by 'Base 3'. Earned Run Average definitely isn't given in base three. It is a simple calculation of (Earned Runs Allowed/Innings Pitches) x 9. The resulting stat can be from 0.00 to ∞, and plenty of pitchers have an ERA higher than 2.xx. Basic example, if a pitcher gave up 1 run, finished the inning, and then was benched, they would have an ERA of (1/1)x9=9.00.
It's never that easy to source something pointing out the error of a premise, but by way of example, if you go to Baseball Reference's career ERA stats there are only 170 eligible pitchers listed with an ERA less than 3, and at number 999, Larry Brown is sitting there at 4.200.
The intention is very straight forward, essentially to extrapolate a pitcher's performance over 9 innings. It was nowhere near the SABERMetrics of today, but back in the early days, stats like ERA and AVG were kinds of statistical tools available for player evaluation, which was just as important then if not as advanced. To be sure, there was a whole hodgepodge of alternatives floated about in the late 19th century for measuring the success of pitchers, such as one example which took win-loss percentage, times batting average, times fielding average. Another wonky stat which was floated was closer to ERA, but normalized it to at-bats (1884's leader was Tim Keefe at 0.0362).
ERA was not a premier stat early on though in large part due to the fact that most pitchers did pitch nine innings a game, so there wasn't as much need to extrapolate like that, and many simply opined that Win-Loss was all you needed. A big proponent of this was the father of baseball stats and journalism, Henry Chadwick. Although he had come up with the statistic himself back in 1867, he apparently detested how Earned Runs were measured - (stolen bases being credited as earned instead of unearned. He had created it as an offensive stat, not a pitching stat!) - and his influence helped to keep Earned Runs from gaining too much traction to measure pitchers until his death in 1908.
Within a few years, John Heydler, a leader in the newer generation of sports writers, started to push Earned Runs successfully as an official stat, and in deference to the increasing use of relievers, it was by 1912 standardized in the current form of average over 9 innings to allow comparisons regardless of complete games, early benching, or relief appearances. It was quite popular, and remained the dominant pitching measure for the the rest of the century (although lets be honest, WHIP ftw!).
Playing for Keeps by Warren Goldstein is a decent early history of the game.
The Numbers Game: Baseball's Lifelong Fascination with Statistics by Alan Schwarz touches on the origin of a number of stat types.
He answered above is super interesting as a baseball fan, but I think misses the point of the question. I believe the OP is asking why you see pitchers recorded as having pitched 5.2 innings, or 3.1. And the answer is it is tracking the outs recorded. So if you start the 5th inning and get 2 outs then are removed you are credited with 5.2 innings pitched. Any batters you left on base that disnt get there via error/fielders choice would still count toward your runs for the game. This is also why you sometimes see pitchers stat lines saying something like 4+ innings; they started the 5th but did not record an out.