Its a dangerous thing to "trust" someone when they'll have that level of control over your future so people use government records and google to know if they could possibly have an issue.
This isn't a privacy issue its a minimizing risk issue.
If you give people better options for minimizing risk then google results will matter less and less. For instance I'd be happy to hire a convict if I could a) pay him less than I normally would (initially) and b) fire him at any time for any reason.
Same for renting. If we draw up a contract with a no-drugs no-weapons clause and allowed me to inspect the place once a month for the first year or so I'd be happy to rent to him.
But these things sensible precautions are illegal. There may be good reasons because of them "evil corporations" blah blah. Sensible or not that's reality - so people understandably discriminate against people based on google results. Its a fairly understandable precaution given the amount of risk involved due to regulation.
Therefore the fix isn't to make google "forget" things - that only increases risk dramatically. The fix is to make these relationships less risky by allowing business owners and landlords more control over their own personal.