30% the other team never touches the ball? This isn't they win overall, it's they win on the first drive that many times. It's not the majority, but it isn't an insignificant number either.
I agree that 15 mins is normally too much, I just think both teams should have a reasonable opportunity to score.
If it was close to or over a majority, I might accept there's a problem. But that stat means 70% of the time both teams get to play offense at least once; in the sudden-death format, I think that's acceptable.
Both teams do have a reasonable opportunity to score. Force a punt or get a turnover. If your defense can't do its job, that's too bad.

Sudden Death is the problem.
Get rid of it. Then you don't have to worry about who gets the ball when.
5 mins is enough time to run several hurry up offenses and that how over time should be played. One Time Out.
It doesn't need to be an entire other quarter. This forces the teams to pass the ball more often than run...or if they feel they are a superior run team...bleed the clock and keep the other team off the field while you score.
Either way would make for incredibly fast finishes and amazing heroics
5 minutes? Single drives can easily take 5 minutes.
And what's more heroic than the knowledge that one big play is all it takes to lose - or win?