Lord Tyrie, who left the competition regulator last summer, said the authority had been "too slow to react" to criticisms about testing providers.
Holidaymakers faced high prices and bad service from many of the 400 approved test firms listed on the government’s website.
Tests cost about £75 on average, but prices can reach hundreds.
Lord Tyrie told the BBC that the CMA "could and should have been better prepared".
"It should either be acting already directly using existing powers. Or if deemed inadequate for the job, it should be advising the government on how to obtain a quick remedy, whether by legislation or by other means," he said.
"This advice should already be with the government."
Lord Tyrie, who spent two years as chair of the CMA, suggested the role limited him from leading a more aggressive campaign for change.
He told the broadcaster: "The CMA acted much more quickly to quell price-gouging on hand sanitiser and other Covid-related products 18 months ago. Far from building on this success, boldness appears to have taken a back seat."
It comes after the government announced the price of NHS travel tests will be reduced.
The government said UK holidaymakers will benefit from cheaper travel testing packages.
Health and Social Care Secretary Sajid Javid has also called for an urgent high-level review from the CMA to address exploitative behaviour in the private testing market and crackdown on excessive pricing or misleading claims.
Mr Javid said: “I know how much people have looked forward to their summer holidays and that the cost of PCR testing can be a barrier to that.
“That is why I am determined to protect consumers and hardworking families from exploitative practices and ensure high-quality tests are available at a reasonable price.”
The CMA said it would explore whether there are pricing and reliability problems.
While the CMA carries out this review, it will provide ongoing support to the Department of Health and Social Care (DHSC).
A CMA spokesperson said: “These comments [Lord Tyrie’s] are inaccurate. The CMA gave advice to DHSC officials about the PCR testing market in April and we gave further advice to the Secretary of State last week.
“We continue to work closely with DHSC in reviewing the market and, as we have made clear, we will not hesitate to take enforcement action ourselves if there are breaches of consumer law.
“Since the start of the pandemic, the CMA has secured hundreds of millions of pounds in refunds for people who have been treated unfairly, in sectors including holiday accommodation and package travel.”