Having been part of an org that asked for volunteers, we did it for two reasons:

First, to try to have a pool of repeat volunteers who we could rely on. In any other context, we would call these people "employees" but we ran on such a shoestring we couldn't pay everyone for all of their labor. Sucks, but such is the nature of the non-profit world outside of the glossy donor magazines. Most of these generous souls were recruited from friends and family or were people who'd received services from us in the past.

Second, is for advertising and to drum up support. People are far more likely to donate to a group they've volunteered with, even once, in the past as they feel a sense of ownership over the organization's outcome. Those "bring a bunch of six-figure salary folks to a group to pack boxes for a few hours?" They tended to result in some of the best donations we received for the whole year.

This is actually a double-edged sword, at least in the USA. You can’t have volunteers and employees do the same work, even at a charity. If you do, you fall afoul of federal wage laws. So you aren’t allowed, for example, to have a mixture of paid and volunteer workers acting as docents at a museum to help with staffing challenges, or have a mix of volunteer and paid workers work the register.

Sorry, I was unclear; I meant we would love to hire all of these amazing people who come back to volunteer week after week, but we can't afford it so we rely on donated labor.

The (very small number of) paid staff jobs and volunteer roles definitely don't overlap. First because our lawyer agrees with you, and second because we feel like that would be extremely rude to have one person be paid while another person is doing the labor for free. If one person is getting paid to do the role, everybody doing that role should be getting paid.

We try to do right by our volunteers. For example, when we have a new paid opening, we tell our volunteers about it first. And sometimes someone donates some money with a note of "do something nice for the volunteers, if you can," so we'll throw a party for our amazing volunteers or buy a nicer lunch (everybody gets fed but sometimes it's boxed sandwiches with chips and other days it's a full taco spread from a local restaurant)..

Free Staff? Is there another term for this that's proper and also doesn't imply compensation?

Is the word you're looking for "volunteer"?

"volunteer staff" is probably close.

"volunteer" by itself is not, because it implies nothing about it being long-term.

Neither one really gets the connotation right.

Repeat volunteer. Long term volunteer.

I'd ask in response if you've ever seen an organization asking for volunteers that isn't also asking for money.

If you can volunteer in addition to sending money, depending on the org they will have some shitwork to do that is unpleasant/monotonous (would cost money to get someone to do) and low-skilled (you probably won't screw it up, and they don't have to waste skilled staff on it - and the skilled staff hate doing it even more than you will.)