Looking for the name of an on-base stat that I am sure exists

I would guess you're referring to On Base Percentage (https://www.baseball-reference.com/bullpen/On_base_percentage).

On base percentage (OBP) includes all methods of getting on base, but does not exclude home runs. All methods of reaching base safely are included except for errors, reached on a fielder's choice, or dropped third strike.

The only major hitting stat that excludes home runs is Batting Average on Balls In Play (BABIP), which also would not include walks or HBPs.

The only area I would imagine a statistician might be interested in times on base (but not home runs) would be in evaluating baserunning, perhaps to use as a denominator; I do not believe there is a named statistic for that, though, they would simply use the OBP numerator and subtract home runs directly.