I don’t believe what you have cited is the Constitution. You are citing the Senate rules, and it does not say “immediately” it says they have to “receive a message” and “appoint managers”.
Those rules can be changed on a simple majority vote.
The laws for impeachment:
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https://www.govinfo.gov/content/pkg/CDOC-99sdoc33/html/CDOC-99sdoc33.htm
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are statutory requirements vested by the Constitution like any other Federal Law until the Supreme Court throws them out or amends them.