Not a fan of pitch counts, innings management, or any of the other ways teams are trying to keep pitchers healthy these days. Throwing a baseball overhand at speeds in excess of 90 MPH is an unnatural motion and is and unto itself asking for an injury. Some players just seem to have a more natural aversion to injuries (see Ripken, Cal) while others are an injury waiting to happen (see Buxton, Byron).
Today's players are in better athletic condition than players of the past, but seem to wind up hurt more often. I think the weight room has probably hurt as many pitchers has it has helped. Pitchers from the past relied more on conditioning and stretching than weightlifting, pitch counts and innings management programs. They also do more baseball related training in the off-season, and you often hear of players getting injured while doing so.
Ober did have an elbow injury in 2019 and missed 2 months, but his recent injuries have been hip and groin strains. Flexibility issues?
My answer to the OP's question is yes, the innings do matter. They matter to getting this team to the post-season, so keep sending him out there. If any limitations are to be applied, go to a six man rotation or a BP game during long stretches of games with no days off, but I don't think that would help the players or the team.