Ok, in a MLB 2018 season I'm playing on the side, eight position players have pitched through 18 games (excluding Ohtana) for a total of 10 2/3rds innings. That seems a lot. However, according to this article, the number of times a position player pitches a season is actually way up over historical averages. In 2016, 26 position players pitched.
https://www.sbnation.com/mlb/2017/8/...for-every-team
While bullpen ERA doesn't look too bad across the league, many teams seem to have mostly tired bullpens.
In my fictional league with 28-man rosters, only one position pitched has pitched over 2/3rds of an inning. That league is 148 games into the season.