Should those be admitted to the Lord's Supper who show by what they profess and how they live that they are unbelieving and ungodly?

No, that would dishonor God's covenant and bring down God's wrath upon the entire congregation.¹ Therefore, according to the instruction of Christ and his apostles, the Christian church is duty-bound to exclude such people, by the official use of the keys of the kingdom, until they reform their lives.