Actually, tithing is addressed in part in the New Testament:
“Woe to you, teachers of the law and Pharisees, you hypocrites! You give a tenth of your spices—mint, dill and cummin. But you have neglected the more important matters of the law—justice, mercy and faithfulness. You should have practiced the latter, without neglecting the former.”
Mat 23:23
The larger lesson is that the Pharisees were meticulous about tithing, even to the minutia of the spices they received, but failed in the matter of righteousness in character.
The smaller issue often overlooked here is that Jesus is also saying to them “do not neglect to continue in your tithes” even with the expectation of holy and righteous living.
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.