Your title asks what thickens an egg yolk. Your description sounds like you want to replace them in creme brulee.
Egg yolks have quite a bit of protein, though not as much as the whites. When cooked, these liquid proteins unravel and tangle up with liquid forming a gel. If cooking continues they tighten up and squeeze the liquid out making soggy scrambled eggs.
Egg yolks don't have much sulfur (the sulphur is in the whites) and bacteria isn't a risk as they are sufficiently cooked in a custard. Not overheating them is not difficult.
I submit that the premise of your question is flawed. Creme brulee is a delicate, but very rich, custard and as with most custards it's texture is defined by egg cooked with high fat dairy. You may be able to produce a firm gel with a polysaccharide but it won't have the texture of creme brulee.
Additionally, the reasons you listed for wanting to avoid egg yolks are not a concern. Little sulphur in yolks, cooked to avoid bacteria, use a water bath to avoid over cooking, etc. If your objection is nutritional consider that the fat in the egg yolks is dwarfed by the cream and sugar in the rest of the recipe.