I would change the institutionalism of home teaching. Strong families with a father in the home don’t need a monthly visit to satisfy the bishop’s eyes. He sees them in church and they may even be his counselor or scoutmaster.
Instead, treat home teaching as an outreach. Don’t have the presidency just assign all families by inspiration (or desperation), but counsel as a quorum, ask for volunteers to visit the widows or the fatherless. Get rid of the monthly scorecard. Use a quarterly quorum meeting to counsel together and make necessary assignments and then use a quorum meeting at the end of the quarter to return and report. Don’t visit active families with active fathers in the home- save that effort for those who can really benefit. If an active quorum member does need some help at home he can ask for it in the quorum meeting.
Someone may say that such a strategic, surgical approach to home teaching doesn’t allow the Church to accurately determine a percentage due to the quarterly list changing based on the immediate needs of the ward. Also, not everyone in the Church would be visited and figure into the quarterly percentages. Exactly my point. It changes from a guilty obligation to sincere service with only God keeping record.