Sauce thickener for large batch

The method I was taught doesn't care about the absolute amounts: First add small amount of the sauce/soup/whatever you want to thicken to the 'powder type' thickener and stir, adding more sauce until it's a very thick liquid without clumps. Only then pour the liquefied thickener with sauce into the 'bulk' of sauce. A very modest amount of stirring will distribute and dissolve smoothly and once it heats up and starts absorbing more water the thickening process will begin for good.

If a lot of the thickener stuck to the bowl in which you were mixing it with sauce, transfer more sauce from the pot to the bowl and dissolve whatever stuck to the walls, then pour back into the pot.

While there are many thickeners, none is easier to apply than tapioca starch. Other starches work the same as tapioca, with minimal differences in the final texture. All other thickeners are harder/more tedious to use, and none is as universally suited to different applications as starch is.

There are no liquid thickeners, that would make no sense - thickeners by definition swell up when coming in contact with water. Compounds that are liquid at room temperature don't swell up when mixed with water, and any mixture of a solid with a liquid will already be swelled, not being able to thicken your sauce further.

I often with use Guar gum or sometimes Xanthan gum to thicken sauces. If the sauce is to be served warm or hot, I will go with the Xanthan gum, but I've found them to work similarly enough in most cases.