The truth is that the town doesn't gain information without lynching
Not true. Because of the difference in knowledge between the mafia and townspeople, the voting patterns for the mafia will always tend to be less random and to vote as a bloc.

Because of this, each lynching is more likely than not to vote for a townsperson, because anyone who bandwagons is almost certainly doing so with an existing mafia bloc vote (who obviously aren't going to vote for a mafia).

The absolute lack of certainly that the townspeople have about who is or isn't mafia makes them more vulnerable initially, but it's also a strength because, every townsperson should avoid acting in co-ordination with others, unless and until there is clear evidence of collusion between certain people. Anyone who does collude with others will either be mafia, or unwittingly working with them to lynch innnocent townspeople.

So the best method is to split the vote inconclusively and not lynch anyone, because besides leaving you with an extra townsperson, giving you more rounds to work with, the results will effectively divide up the town into smaller groups and individuals, which you can then track to see if they continue to try to vote as a bloc. You'll also be able to rule out any isolated individuals that a consistent voting bloc goes for as being mafia.