Thermals make it easier to solder components onto your board. Less metal means less heat drawn away from the joint. I find that thermals make soldering on voltage regulators way easier.
Is there an advantage for other components?
Wouldn't that decrease the amount of current that can go to that component?
An advantage to anything that could be hard to solder. If you are hand soldering it isn't a big deal, but really helps for mass production.
Less current, yes, but it can surely still handle more current than your component can. A thermal might bring it from 8A to 4A, but your voltage regulator can't handle more than 1A anyway so no biggie.
I believe those are "thermals." If you turn off "thermals" then you'll probably get what you want. But your board will possibly have other problems as it's actually being used.
Best is to make the line size of the polygon pour fairly thick. The draw-back of that is that then it won't actually snake its way through narrow openings.
Turned it on and off, didn't work. Would I have to recreate the polygon for the thermals to disable?
I think you can just de-wire than re-wire the polygon and it'll work. I can't remember the exact terminology though, and too lazy to open Eagle to double-check.