Their idea was all your money is really God's; forget all that rendering unto caesar stuff and if you're richer or poorer it's just God micromanaging your finances.
Among the more bizarre ideas I saw was that no person can decide where the tithe is to be used. I'm not sure how that works since churches are built, occupied and run by people.
People point to Leviticus yet ignore other parts that include diet and specific penalties for behaviors including stoning.
In the quote you offer from Matthew, He's speaking to Pharisees, Jews who naturally would be tithing. He is holding them to the standard of Jewish Law to highlight their hypocrisy. You seem to suggest Christians are not Christians if they aren't good Jews first and foremost.
In the OT, tithing supported the temple priests, distributions to the needy, and for maintenance of the temple. These needs are consistent within a balanced and functioning NT church.