> No matter what point n we choose, the objection "why is one day earlier any better" seems pretty persuasive. Therefore, by induction

That's not persuasive at all. It's not just not "bulletproof", it's blatantly wrong. Also you can make the same argument in the other direction.

> Indeed the argument doesn't even need to be correct for my point. My point is that nothing about that argument requires any religious belief whatsoever.

They wanted someone to give a plausible argument that isn't religious.

> no reason to believe that secular arguments against abortion can't exist

I care about the merits of positions that people actually have, not theoretical positions.

And in the general case, if nobody can be found that has a simple position, that is a reason to believe it's not a coherent position.