Without seeing the loaves or knowing your exact procedures, it's hard to guess. My money would be on too dry a dough. Could also be inactive starter, too long a bake, not a hot enough bake, not enough rise time/temperature. So I'll just talk about how I make sourdough in the hopes it'll help. These are techniques I've learned from various books (most notably Reinhart's The Bread Baker's Apprentice) and experiences.
If I did it right, you can click any of the thumbnails for full size images.
First order of business, of course, is having a nice, active starter. I build my final starter from the barm the day before. Depending on my schedule, I either leave it at room temp all night or feed it earlier in the day and pop it in the fridge. It then comes out to warm up a few hours before I bake. It should show strong signs of life. This starter was made with nothing but flour and water--no grapes, no commercial yeast, no juice.
Then comes mixing and kneading. I like to mix the starter with the water first, allowing the starter to break up and dissolve a bit. This ensures good mixing with the final dough. Then I add the flour and salt, mix to a coarse ball...
...and switch to the dough hook for 5 minutes or so. The dough should be stickier than you think it should be. Turn it out onto a floured surface for 4-5 minutes of final kneading by hand to adjust the flour. It should be smooth and supple, still be a bit tacky, and pass the windowpane test. When you stretch it to form a boule, the outside should be smooth and not tear.
Let it rise 3-4 hours at room temp or slightly warmer (a closed oven with a bowl of hot water works well). Then shape and proof for another 3-4 hours.
I start with a very hot oven. 500-550F for artisan loaves. If using a stone, get it in there 45-60 minutes before baking, that way the stone and oven get a good heat soak. Then when the loaves go in, you want to create steam to allow the bread to rise before the crust hardens. A steam pan is a good idea, and I also spray down the walls of the oven with a squirt bottle. Be careful of squirting the light bulb--it will shatter. Don't ask me how I know. After a few squirts, turn the heat down to your final baking temp. I usually go with 450F.
Don't forget to slash them, too. That helps them expand and prevents blowouts. I had some issues getting these guys onto the stone (not my finest hour), but they were still outstanding. You can see the crumb is open and light, but still gelatinized and chewy. Perfect, in my book.
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.
Best Answer
I have been making sourdough bread with a few kinds of flour. In my experience, bread flour (higher protein) really does make a difference in many qualities of bread that you look for -- texture, rise, chew, etc. However it is incremental, maybe like 10-25% better, so it should still be possible to create good bread with lower protein flour.
I can give a few tips for your starter that I recently learned. The starter should be at least doubling if not tripling or quadrupling in volume each time you feed it. When I feed my starter I add 1/3 starter to 1/3 fresh flour and water. So for example if I have 100g of starter, each time I feed it I take 33g of that starter (discard the other 67g) and add it to 33g of each flour and water. After 2 or 3 of these feedings it at least triples in volume each time. If it is sluggish, I make sure it is in a warm spot, ideally about 85F. There can be a big difference in activity between this and a cool room temp of 65F.
Also the starter should be peaking in activity at the time it is added to the dough. For me that's usually about 3-4 hours after feeding but it depends. If it has fully deflated before you use it the activity will be less.
Something I have started doing is overnight fermentations of the dough, usually about 18 hours, at room temp. This results in bread that is much more sour and has a wetter, stickier texture that I like, so might not be for everyone. However it makes it easier to get a good rise. My dough will double or triple in volume overnight and there is not much harm in having it sit for a few more or less hours. The activity and strength of the starter matters a bit less because it sits for so long. When I'm ready to bake, I will "stretch and fold" the dough and it usually rises back up within an hour or two and is ready to bake.