Yeah, most leagues I play in do the following.
Overall Record
H2H
Total Points
Sucks, but from my experience, it's pretty normal.

I don't even necessarily think it sucks.
The logical extension of complaining about H2H for a tie-breaker is complaining about H2H (record) to establish standings which resulted in tie.
The logical extension of complaining about H2H is points league.
So long as it's your rule, no injustice has been done.
This time of year, all sorts of rules and notions are promulgated, in the name of 'fairness' and 'minimizing luck'. Most of them are ill-conceived and ill-advised.
We are going to put it to a vote as this is the second yr in a row this happened.
We are going to put it to a vote as this is the second yr in a row our playoff seeding was decided by the rules we put forward.
Careful what you wish for.
I personally like having H2H as a seeding tie-breaker. Throughout the season, when I play the other top teams in the league, I know that those matchups have a potential impact on playoff seeding tie-breakers. Other teams know this, as well. People keep track of who they beat and who they lost to, and also who else beat the other guy. This generates greater interest within the league during the regular season.
He got the playoff bid because he beat me twice head to head.
When you lost to this other team twice during the season, did the playoff seeding implications not come to mind?