“House has to seat but House doesn’t have to keep.”
It’s more than that. The Supreme Court has ruled that the house cannot exclude members under the Constitution. This is an excerpt from Wikipedia describing Powell v. McCormick. I think the ruling makes complete sense as the Constitution sets out certain requirements for being a congress critter. Not being a criminal is NOT a constitutional requirement and the people elected him.
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The full House refused to seat him until the completion of the investigation. Powell urged his supporters to “keep the faith, baby,” while the investigation was underway. On March 1, the House voted 307 to 116 to exclude him, despite the recommendation of the Select Committee. Powell said, “On this day, the day of March, in my opinion, is the end of the United States of America as the land of the free and the home of the brave.”[35]
Powell won the Special Election to fill the vacancy caused by his exclusion, receiving 86% of the vote.[36] But he did not take his seat, as he was filing a separate suit. He sued in Powell v. McCormack to retain his seat. In November 1968, Powell was re-elected. On January 3, 1969, he was seated as a member of the 91st Congress, but he was fined $25,000 and denied seniority.[37] In June 1969, in Powell v. McCormack, the Supreme Court of the United States ruled that the House had acted unconstitutionally when it excluded Powell, as he had been duly elected by his constituents.[38]