The Scot, who won the second of her Olympic gold medals in Tokyo last year alongside Dame Laura Kenny in the Madison, said the UCI had "chosen to delay action until it became sadly personal for one rider".
Bridges, 21, was due to compete at the British national omnium championships earlier this month having met British Cycling's eligibility standards, but was blocked by the UCI at the 11th hour before the national governing body announced a review of its policies.
On the eve of the Nations Cup in Glasgow this weekend, Archibald issued a statement which said: "It is my opinion that the international governing bodies of several sports have let down transgender athletes, in particular transgender women, with their inclusion policies.
"These policies have put the athletes, their involvement in sport, and their personal lives under intense scrutiny when all the athletes have done is follow the rules and enter a category they were encouraged to enter.
"I, too, feel let down by these policies. I feel let down by the International Olympic Committee who tell me there should be no assumed advantage for an athlete with a gender identity different to their sex.
"I read this and hear that my world titles, my Olympic medals, and the champions jerseys I have at home, were all won in a category of people who simply don't try as hard as the men. That losing to male androgenisation is not about biology, but mindset. They are wrong.
"The retained advantage of people who have gone through male puberty in strength, stamina, and physique, with or without testosterone suppression, has been well documented.
"Cycling's global governing body, by its president's own admission, knows this. But they chose to delay action until it became sadly personal for one rider. That wasn't fair."
Bridges, who was in British Cycling's senior academy in 2019, came out as a transgender woman in October 2020. She continued to compete as a male during her transition, and in February won the men's points race at the British Universities' championships.
She had met British Cycling's previous requirement - that riders in the female category have had testosterone levels below five nanomoles per litre for a 12-month period prior to competition - in time to register for the championships in Derby, but the UCI did not grant a switch in licence.
Bridges issued a statement after that decision saying she had been "harassed and demonised" amid the debate, while her mother Sandy used social media on Wednesday to say "the damage caused to Em's wellbeing is inexcusable".
At the time, UCI president David Lappartient told the BBC he was concerned regulations do not go far enough to protect the integrity of competition.
A week after the championships, British Cycling suspended its transgender policy and announced a review, saying it needed to "find a better answer".
Bridges had been due to compete in Derby against a field that included Kenny, though Archibald was out with a back injury at the time.
Archibald said she had sent a message to Bridges prior to issuing her statement.
"It felt like I couldn't continue making no comment when it's an issue which has been put to us so consistently for the past I don't know how long," Archibald said.
"There's only so many times you can say 'I'm not the person with all the answers. It's not my job, or my team-mates' jobs, to write policy. We just want to ride our bikes'."