Natural gluten in starter - good or bad?

It makes no difference.

1.)  The starter's there for the dual purposes of providing leavening and adding taste.  You'd never be adding enough gluten from "stirred" starter to make much of a difference to total gluten content.

2.)  Gluten's developing even if you don't stir the starter mixture anyway.  

So don't bother; or bother, if you like, but it won't make much of a difference either way.

stirring is to distribute food and nutrients to the bacteria and yeast growing in the starter.  

Starters can be grown without gluten, they just don't expand like wheat flours do. (Wheat gluten being used as the "norm.)  One looks for other signs of the starter's ripeness.  

Gluten will also be broken down and destroyed after a period of time by the activity and waste products of the bacteria and yeast.  That means that an underfed starter with exhausted gluten structure may not rise but still contain yeast and bacteria and enough food for some of them to survive.  

Stirring and evaluating the effects of starter acid on gluten might have the purpose of making a judgement on how ripe or far gone or how useful a starter can be.  If you use that method of judging your starter,  gluten will start out tight and with time soften and be smooth eventually turning liquid, separating and getting stringy, falling apart completely.  So will the yeast numbers drop during this breaking down process.  Bacteria numbers will increase.  That is fine and good in building a separate starter for flavour and is often pushed to its limits before adding into a recipe and granted, it won't rise much unless fresh flour is introduced for more population growth but, but don't treat your mother starter this way if it is a wheat starter.

You can do an interesting experiment...(use discard for the fun)  

Take two piles of flour.  Exactly 100g and 50g.  Blend the 100g pile with 50g of water to make a dough. Blend the 50g with 100g of ripe sourdough (the sourdough is 100% hydration and made with the same flour as the experiment.)  

You have two dough balls of 150g each.  One just flour and water, the other with sourdough.  Let them stand for 30 minutes or so and then knead them to make sure you've developed the gluten, use wet hands if need be but don't add flour.  

It is important to be exact in the measurements of the flour as the next step is to gently wash out the solids in the dough to reveal what is left, namely gluten.  

To do this place a dough ball in a bowl of water and gently knead the ball to reveal a ball of stringy gluten, finish off under a gentle stream of water until a grey ball is free of loose flour and water is clear.  Now do it with the sourdough ball, squeeze them as dry as possible and then weigh and compare.  What are your results?

Nobody could explain it better!

Karin