I get where you are coming from. You are correct they did fact check Trump's outrageous and blatant lies about abortion and eating dogs and cats, both of which had recently been debunked. They could have let them slide because one hopes that most people knew both had recently been proven false... many times over.
Networks including CNN and FOX fact check usually immediately following a debate, town hall or rally. Getting multiple points of view about what was fact and what wasn't is not difficult.
People who are interested in knowing whether candidates lied or not during a debate, rally or town hall will look further... or they should. What is more, most people know that politicians exaggerate and sometimes lie especially during election campaigns and do not accept everything they say on face value.
I don't think they were proven false. As Vance said, he had heard these claims from constituents, and the abortion one I saw proven correct, but I don't really recall it.
However, regardless, they fact checked one and not the other. I imagine the debate would have gone much differently with moderators who were fair. Imagine this:
"Madame Vice President, President Trump did not refer to nazis and white supremacists as "very fine people". He specifically stated he was referring to the folks in the statue debate, and condemned the nazis and supremacists. This has been known for years"
"Madame VP, we actually DO have our military in active zones over the world, your statement is incorrect"
"Madame VP, Donald Trump has repeatedly said he would not enforce a federal abortion ban, so that is a lie"
"Madame VP, Donald Trump has, on multiple occasions, denounced Project 2025 and says he wants no part of it, so that is a lie".
She would not have been able to operate with confidence and impunity. She would have stumbled greatly.
