https://www.govinfo.gov/content/pkg/CDOC-99sdoc33/html/CDOC-99sdoc33.htm
IV. Sequence of events at the beginning of a trial...................9
1. First a message(s) from the House of Representatives
is received containing the information that the House
has voted impeachment, adopted articles, and appointed
managers. The Senate then adopts an order informing the
House when it is ready to receive the managers to
present the articles of impeachment.................... 9
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.