The amount of radiation is just too high.
The fact that it interacts with stuff is actually what makes it dangerous. If it went through every thing it would not interact with the worker either and would not pose a thread to them.
Btw. there is a type of radiation that is practically like that: neutrinos. Impossible to shield against, but also completely harmless if you are not standing right next to a super nova.
There was actually a fun what-if based on that: https://what-if.xkcd.com/73/
I'm picturing a sheet of paper with bullet holes through it.
Or is there a better analogy of what is going on?
You can think of it as roughly analogous to the excitation of atoms leading to emission of a photon in the visible range. It doesn’t help to wear a pair of goggles that will absorb all of the light you’re hoping to block, only to have your goggles glow brightly as a result. You’d rightly conclude that you need much thicker goggles, and the same is true here.
No holes needed, and a lot of the original gamma radiation is being absorbed and subsequently emitted, which defeats the point. There are also complications of a substance being exposed to some forms of radiation (primarily neutron) becoming radioactive themselves. It wouldn’t be visible, but it wouldn’t be good either.
A note that shielding, in addition to considering total flux, has to consider the type of radiation. Gamma (very high energy photon) is very penetrating, beta (high energy electron) is less so, and alpha (a helium nucleus) hardly penetrates at all. You also have to consider, in the case of a molten core, other contaminants and radionuclides actually getting on you or in you. Alpha emitters are pretty harmless outside of you, where your skin stops them cold. If you eat or breath them in however, they’re deveststing. So your concern with alpha emitters tends to be covering yourself and wearing a respirator, while that won’t even help a little with gamma.
The equivalent in lead would probably manifest in some way, perhaps a visible patina on the surface? or, as a differential in decay products. you could call that 'damage' if you wanted to.