Giving to the Church is not giving to the Needy. Ditto for donations to Colleges, Hospitals, Libraries, Museums and other institutions.
The Verses you quote discuss "Almsgiving" which is the donation of surplus money to the poor and destitute, to widows, to orphans, the sick, etc. That is, humanitarian relief, and a just (PRIVATE!) redistribution of the goods of the world.
Tithing, and donating to Institutions is not "Almsigiving" but "Benefaction". There is nothing wrong with a Benefactor being known, so that people might be both grateful to him, and also pray for him to God. But the two are distinct. Almsgiving is a form of the virtue of charity and is something voluntary arising out of the free will without any compulsion or obligation upon our part towards the person who benefits. It is also a gift of money, goods, or time that ultimately goes to a specific person. Benefaction is an obligtion towards society in social justice for the holders of excess goods. Benefaction benefits an Instutition, that serves the public, not specific persons in need.
Look at it this way. Andrew Carnegie was very rich. He had an obligation to society to uses the riches it had permitted him to acquire to benefit society as a whole - hence his libraries and museums and educational and research instutitions. He also had an obligation to assist the poor out of his abundance, but not an obligation to assist every poor person, so that what he did for a specific one might be an act of voluntary charity instead of something compelled. It is these acts of charity which should be secret, not our donations to societal instutions, including the Church.
Thank you.
Thithing (or any donation to a church) is an "act of rightousness" and should be done in secret. Look again.