> The police are asking you questions to "get to the bottom of it" and they encounter people every day who do think they can lie to get out of a crime;

I don't even think they are all that interested in getting to the truth of the matter, they are mostly concerned with getting an arrest and conviction. If it'll be easier to throw you behind bars than to find and arrest the dangerous person who actually committed the crime they aren't all going to choose more work and risk "officer safety" when they can just take you and call it a day. Especially not if they're already prejudiced against you or you bruised their fragile ego somehow.