How do I make my sourdough bread dough stretchy and not have it tear?

Your recipe is to blame.

Your dough is stiff and tears because it isn't wet enough. While different types of flour do absorb different quantities of water, I think most of the issues you're experiencing are due to the ambiguousness of the recipe and instructions you're using. Most sourdough recipes yield dough around 80% hydration (the weight of water/the weight of flour). The recipe you've cited relies on the starter providing all the hydration for the dough. Doing the math, following the feeding instructions, you would have a starter that is 55% hydration. Adding all the flour called for in the dough recipe, you would end up with a dough that is about 45% hydration... stiffer than bagels (50-55%).

While it is possible to make dough that doesn't have any added water, it is not common, especially with such a stiff starter. Also, the volume of flour would be much smaller than the volume of starter.

Try using a tried and true recipe that doesn't rely on the starter to provide all of the hydration.

Don't worry so much about gluten production and how sugar or fat will affect it. Small amounts of these enrichments, like those in the recipe you linked to, will soften the interior of the bread and help the crust brown nicely. Just look at brioche if you want an example of how far you can push enrichment and still get a well structured bread.

I blame the acid.

Sourdough starters can get very acidic and this breaks up the gluten and makes for very short dough, and dense bread.

I've had a lot of success using less starter and let it rise in the fridge. You'll get all the flavor without letting to much acid break things up.

In general, your expectations of sourdough should not be the same as for other artisan bread. Sourdough bread does not have as open a texture. The more sour it gets the stickier and the smaller the crumb.

When all else fails you can still form a boule. It will look funny as it will flatten as it proofs but will still taste great.

I agree with the "isn't wet enough" answer by Didgeridrew, but I don't think the problem is the recipe. You say

I kneaded it for about 1/2 hour to 45 minutes. I tried to avoid adding too much flour to keep the water percentage up, but that's hard to do when you are kneading on a wooden board and the dough keeps sticking.

That's your problem right there: you kept adding flour. If you are doing it for 45 minutes, you added a lot of it.

I know that it's very unusual to start kneading wet doughs, but this is indeed how it's supposed to go. It will stick. Let it stick. A really wet dough will smear all over your hands and the board; this doesn't matter. Just keep kneading. Don't add any flour at all. When the gluten is ready, it will pull everything back together by itself. Once you get accustomed to it, you'll be able to handle really wet dough - I've done 90%, I didn't have a board there, but had to pour it from one hand into the other, it flowed like lava for the first 15 minutes.

And another thing: decide what hydration percentage you want and stick to it. Measure by weight. The "flour varies by absorbency" part is indeed true, but you have to be experienced in dough handling to know when your dough needs more water and when less. If you aren't, stay with the exact hydration amount. And don't make changes to a recipe; if one doesn't work, find a better one. Changing is much harder.

And a final thing: don't get too hung up on the windowpane test (stretching without tearing). Once you have kneaded a few doughs by hand to the proper stage, you will notice what consistency you need. The windowpane is not completely reliable.