Your issue may be with the strain of the starter - in my
experience, they all have different characteristics. I've used 4 types of sourdough starters over the years. Three came from Cultures for Health, one was a wild one. I didn't keep the wild one going because I didn't like the flavor.
In the Cultures for Health literature, they say that different strains replicate at different rates. Some are naturally faster than others. The rye sourdough starter I had was very slow and I eventually let it go. It also was intensely sour when used with wheat - too sour. I love sour but this made people's teeth hurt! So it was slow to develop, and had a crazy sour result - which I think means that the organisms in that batch had a very high tolerance for their waste products (sour).
You couldn't use that starter faster than it was ready or it hadn't developed properly and your bread wouldn't rise properly - does that make sense? The cultures had their own timing, and a very high tolerance for their sour by-products.
After a couple years of babysitting multiple sourdough starters I realized I wanted an all-purpose sourdough starter.... one that would work for pizza, a nice crusty sourdough bread, English muffins, and a really good rye bread for sandwiches.
In the end, I stuck with Camaldoli Sourdough starter, which is fast rising/replicating (about 4 hours) and can create a wide range of sourness. So at 4 hours, the starter has totally risen and is only very mildly sour. Great for pizza - people don't even realize it's soured dough if I am attentive to the rising process. We like it sour, but this way I can serve it to an assortment of people. When I use it the Camaldoli for bread, I leave it longer and it makes a nice, tart, well-developed sourdough or soured rye bread. I have it separated into a batch I feed with wheat flour, and a batch I feed with rye flour for this purpose. The rye bread recipe I like requires the rye portion of the dough to be all starter.
I'm not an expert on how these cultures work, by any means, but this was both my experience and what I've read up on. So if you want a faster starter, you might need to introduce new organisms to your mix. I believe that would be easiest to do from someone who already has a fast starter. But I love experiments! Please do let us know what your results are in the end.