The sincerity of the belief is a measured by the credibility of the claimant in each specific case. They are assumed to be credible by default unless the other party can show glaring inconsistencies. Even then, the guidance notes that people's beliefs change so challenging such a claim is relatively hard.
A sincerely held religious belief is basically a "get out of jail free" card (figuratively) because of the issues you point out. With the preponderance of evidence standard, all one has to do is convince the judge that their belief is more likely to be sincere than not. If one can prove that they go to church on a semi-regular basis, for example, it's nearly impossible to convince a judge that the belief is insincere.
The thing that's causing issues is that many people who are objecting to the Covid vaccines are largely not anti-vax in general, and their reasons aren't really rooted in the philosophy of religious law around vaccines, but the aggressive and sudden push for these particular ones is tripping all of their mental alarms, because aggressive and sudden intrusions by untrustworthy governments are often used to accomplish evil that is extremely contrary to their religious beliefs. A court probably isn't going to be sympathetic to that argument, but I think it's what's happening.
>Even then, the guidance notes that people's beliefs change so challenging such a claim is relatively hard.
It is not relatively hard, it is (or should be) impossible.
My religion stating that I believe in whatever I want to believe and is subject to change at any time is as valid as any other religion, from the perspective of the government. Assuming we are striving for separation of church and state.
Religion never needs to come up in legislation, just like one's favorite tv shows or music.
Are Catholics sincere if they have sex before marriage? Use birth control? Get a divorce?
Are Jews sincere if they eat non kosher food? Muslims if they eat non Halal and drink alcohol and gamble?
Christians who use Christ to persecute others? Such as gay people who want to get married?