The police had a point, if you can't prove ownership. I can't really fault their response, except that the way they put it might have seemed personal and insulting.
I'm glad you got it back without incident though. "Possession is nine parts" cuts both ways. Yet, I wouldn't commence with a letter writing campaign which could be inviting trouble. Consider: one party admits to "stealing" the bike, the other party hasn't admitted to anything. I'd let sleeping dogs lie.