I'm not sure whether ridicule is the right word for the reactions of other prominent Republicans; disparaging might be closer, but other Republicans are certainly skeptical of contact-tracing efforts. However, the skepticism seems to be predominantly due to implementation details, rather than misgivings regarding its effectiveness.
In particular, Rep. Andy Biggs of Arizona has characterized the contact-tracing provisions as "unconstitutional and crushing government action":
Biggs, the chairman of the Freedom Caucus, took issue with the $12
billion set aside in the legislation for coronavirus contact tracing,
where health workers try to track down individuals who have tested
positive and their contacts.
“Does it allow big tech companies to surveil and trace American
citizens and then turn that accumulated information over to the
government? How will this data be secured, stored, etcetera?” Biggs
said in a floor speech before the vote. “There are many questions that
go unanswered, not the least of these, however, is the question of how
much longer the American people acquiesce to unconstitutional and
crushing government action."
Additionally, a group of Republican Senators has also announced plans to introduce a privacy bill in order to regulate data collected by contact-tracing apps. However, the Senators don't seem to question the effectiveness of contact-tracing as a pandemic mitigation measure - below are comments from Roger Wicker from Missouri, chairman of the Senate Committee on Commerce, Science, and Transportation:
“As the coronavirus continues to take a heavy toll on our economy and
American life, government officials and health-care professionals have
rightly turned to data to help fight this global pandemic,” said
Wicker. “This data has great potential to help us contain the virus
and limit future outbreaks, but we need to ensure that individuals’
personal information is safe from misuse. I am pleased to join
Senators Thune, Moran, and Blackburn in introducing legislation to
address this critical issue.”
Finally, another issue picked upon by some Republicans is the feasibility of rolling out such a scheme:
Rep. Andy Harris, R-Md., said that increasing testing would lead to a
higher number of reported cases and suggested using the number of
hospitalizations or ICU capacity to determine when states could lift
restrictions, instead of the number of reported cases. He suggested
widespread education on how to stop the spread of respiratory
diseases, rather than contact tracing, arguing education could be more
“To train 100,000 people nationwide to institute contact tracing could
take weeks. I have small businesses owners in my state, in my
district, tell me they can’t last weeks," he said.
Skepticism also seems to be prevalent within the voter base - Republican voters are also less likely to be willing to download and use a contact-tracing app, according to a KFF poll published at the end of April - 57% would be willing compared to 81% of Democrats.
Although there is a good deal of skepticism about how data will be protected and citizen's rights will be maintained, there seems to be little to no argument about the effectiveness of contact tracing once the scheme is up and running, and I can't recall any comments made by prominent Republicans similar to those made by Giuliani.