It seems to be that you've convincingly argued that the answer to your own question is no. If there were a “natural” way to define a distribution $δ_p$ for every $p$, then comparing (chartwise) it to the distribution density $δ_p$ would give a smooth density (alternatively: find the smooth density on $M$ so that the integral of the putative distribution $δ_p$ against this density gives $1$ at every $p$); and no such “natural” density exists, so neither can a “natural” distribution $δ_p$. I'm just rephrasing pretty much what you yourself explained: why does it not convince you?
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