What is Bob finds out belatedly that kid as actually Jim's and Mom neglected to ever mention that 'uncertainty' or worse knew but decided Bob has deeper pockets? Many states have decided that once your on the birth certificate your stuck, as otherwise the State suffers having a financial burden.
Yes. Acknowledging paternity (or marriage in a jurisdiction with a conclusive presumption of paternity) creates, and ought to create, obligations which cannot be abandoned easily unilaterally.
It you are unwilling to accept that responsibility it's easy enough not to do those things.
A deeper question might be is it reasonable to be able to take your name off in the first place after affirming it. In a contrived example we may imagine bob discovers he isn't the father while the ink is still wet after being deliberately deceived.
Less contrived examples are apt to be on average more complicated. The uterus isn't an egg timer. Exact date of conception is liable to be fuzzy. The person on the birth cert is more apt to be the long term boyfriend or spouse than the richer fellow. The failure to communicate uncertainty a function of shame not greed. Discovery might be years later after the obligation is deeper than a signature. At that point do you have a right to reject the child you asserted was yours. I don't think so. I think if you asserted that position you own that obligation for your life. Sometimes the law affirms what we as a society hold to be general true and is optimized for the general case not edge cases.
https://en.m.wikipedia.org/wiki/Paternity_fraud#:~:text=If%2....