Is there a real reason not to grate cheese a few days ahead, or is the bagged stuff just worse for other reasons?

Question 1 has a clean answer: it is something added.

Bagged grated cheese is dusted with an anti-caking agent — usually potato starch, cellulose or a similar powder — to stop it welding into a block in the bag. That powder does not melt. In a sauce it stays as fine solids and it also soaks up liquid, which is exactly the grainy texture you have noticed. It is not that the cheese is worse; it is that you are melting cheese plus starch.

The useful consequence: the same bag is completely fine scattered on top of something that goes under a grill, where nothing needs to emulsify. The rule is not "bagged cheese is bad", it is "do not build a sauce out of it".

Your own grated cheese has no such coating, so it will melt properly on day three. That is the good news for your actual plan.

What your own grated cheese loses is mostly moisture, and the mechanism is surface area.

A 200g block has maybe a few square centimetres of exposed face. Grate it and you have created an enormous surface, all of it losing water to the air. Within a couple of days the outside of every shred is dry, which is why it goes hard and why the shreds stick to each other — the surfaces dry, then fuse where they touch.

It also oxidises faster for the same reason, so a strong cheese will taste flatter and a mild one can pick up a slightly stale note. Neither is a safety issue and both are mild compared to the drying.

So the answer to 2 is: mostly moisture, and the fix is a moisture problem, not a preservation problem.

The thing that actually works for grating ahead, and I do this weekly:

Grate and freeze, not grate and refrigerate.

Spread the shreds on a tray, freeze until loose, then bag them. They stay separate, they keep for months instead of days, and they go into a sauce or onto a pizza straight from frozen with no thawing.

The reasons it works: the drying that ruins refrigerated shreds is dramatically slower at freezer temperatures, and freezing does not hurt melting cheese in any way you would notice. Hard cheeses take this best — a cheddar or a gruyère is unchanged. Soft and fresh cheeses do get crumbly, so those stay as blocks.

One caveat: freezing does make cheese slightly more crumbly, so it is worse for eating cold on a board and identical for anything melted. Since you are batch cooking, that is the trade you want.

If you would rather stay in the fridge, the compromise that gets you a few extra days is a proper airtight container plus a piece of kitchen paper on top, changed if it goes damp.

The paper is doing something specific: cheese gives off moisture but also dries at the surface, and what you want is a small humid buffer that is not condensation sitting on the shreds. Sealed tub with nothing in it goes slimy on top and dry underneath.

Still only buys you five or six days. The freezer is the answer if you want a week or more.