Available evidence shows that some prominent Republicans did publicly criticize Trump’s post‑election legal efforts, but there was no broad move by leading congressional Republicans to curtail or repudiate Rudy Giuliani specifically, nor was he pushed off center stage before the January 5, 2021 Georgia runoffs.

Key points:

1. Giuliani stayed the face of Trump’s effort well past January 5. He continued leading strategy sessions and working with allies at the Willard Hotel “command center” in December, and remained central to the broader attempt to overturn the election into early January 2021, including planning around the January 6 certification fight. (en.wikipedia.org)

2. Criticism from within the GOP was scattered and often generalized, not a coordinated congressional repudiation of Giuliani. Some Republicans criticized the legal team or the fraud claims:

   • Former New Jersey governor Chris Christie (not a member of Congress) called Trump’s legal team, including Giuliani, a “national embarrassment” and urged Trump to stop the challenges. (uproxx.com)
   • Sen. Joni Ernst slammed Sidney Powell’s conspiracy claims as “absolutely outrageous” and “offensive,” but framed it as a rebuke of Powell and the broader claims, not a specific move to sideline Giuliani. (iowastartingline.com)
   • Sen. Ben Sasse, in a December 31 statement, denounced the project to overturn the election as a “dangerous ploy” and “swampy politics,” criticizing the fundraising and legal challenges generally, not Giuliani personally. (ottawa.citynews.ca)

3. Congressional leadership did not publicly move to shut Giuliani down. Senate Majority Leader Mitch McConnell acknowledged Biden’s win on December 15 and privately urged senators not to join House objections to Electoral College results, but he did not publicly single out Giuliani or call for his removal from the effort. Meanwhile, figures like Josh Hawley and others still backed objections to the electoral count, aligning with the broader challenge Giuliani was fronting. (en.wikipedia.org)

4. Internal dissatisfaction with Giuliani did not translate into congressional action. Reporting indicates Trump himself and some allies worried Giuliani and other lawyers were “fools” making him look bad, yet Giuliani remained Trump’s point man on the election challenge. These were White House/political complaints, not moves led by congressional Republicans to remove him from center stage. (cnbc.com)

Given that Giuliani remained the public and strategic leader of the post‑election effort through and beyond the Georgia runoffs, and that congressional Republican criticism was limited, fragmented, and not clearly aimed at curtailing Giuliani’s role, the prediction that “leading congressional Republicans” would publicly move to sideline him before January 5, 2021 did not come true.