What takes over the functions of gluten in gluten-free bread mixes?
To summarize the points of Aaronut's answer and provide a framework for answering:


  'Gluten is responsible for elasticity of dough, which is perceived as chewiness','The "rising" in baked goods is essentially just
  stretching of the gluten network', 'Gluten is also exceptionally good
  at both absorbing and retaining moisture'



Dough's rising is catalyzed by the yeast eating sugar and giving off gas, the gas becomes suspended in the gluten network which has body from absorbed moisture and is restrained and stregthened by the salt. The end. 
In the absence of gluten, there is no network for the gas to lift. However, when you add xanthum or guar gum, their qualities as hydrocolloids allow them to swell with moisture and rise with the gas.


Modified rice starch imitates gelatin in some applications, gelatin also produces flex and supports a smoother texture (common complaints about gluten-free breads refer to their grainy texture). 
If you remove the gums and hydrocolloids from the gluten free dough you'll get a sopping wet mess that does nothing; using flax and chia seed will pull it back into shape. Both seeds, when milled will gel andprovide the meringue-type (though not as effective or durable) effect.
Tapioca starch also gives body and accommodates a smoother texture, hence it's addition to the King Arthur's gluten free flour mix.




  'When baking without gluten, you will have to be very precise about all of your measurements', as gluten is more forgiving due to its
  slow-acting nature



Yep. 
In gluten-free baking one must substitute precision for not precision. Over kneading can be fatal especially in recipes with flax and chia.



  'Its coagulation action is actually very similar to that of egg whites... gluten is basically doing the same thing inside of whatever
  you're baking'



Gluten allows you to replicates the function of eggs in some use cases.
Flax and chia eggs as well as Ener-G egg replacer are common vegan replacements to eggs, but as this question is simply about gluten-free baking the answer is to just use an egg; some recipes call for one, some don't. However, there is no likely use case where the functionality of an egg being replicated by gluten would preclude using an egg.



  'Finally, it provides nutritive protein when eaten. Wheat gluten is about 75% protein':



Yep.
Get protein somewhere else. There are 2 grams of protein in a slice of white bread, have an ounce of black beans.



  'In short: Gluten does a lot of things. Keep in mind when doing
  gluten-free baking that a lot of the substitutes only replicate one or
  two of the effects.':



The combination of multiple agents is generally necessary to provide the same functions. If you look at this methodology for high-rising gluten-free dough from King Arthur's, you'll see that the bread flour alone contains a super-fine flour, potato starch and tapioca starch. The recipe calls for xanthum gum and an egg in addition to the tapioca and potato starches. There you already have four ingredients doing the work of the gluten.