The water you use and the flour you use will all influence how well the yeast 'grows'. Tap water containing chlorine may not be ideal. The more processed the flour the less growth you may get. Less processed flours often contain some naturally occurring yeasts.
If you have a few minutes, please try to read this wiki article: wiki article about Sourdough. I think it may explain quite a bit for you.
I just don't think that leaving your 'starter' out on the counter will produce anything BUT a sourdough-like result though it may be subtle. You will cause fermentation to some degree - and more on the counter than if refrigerated. However, not ALL sourdoughs are really tangy/sour. I have tasted a lot lately which were purchased from grocery stores, labelled as sourdough, but with so little tang I might not have known but for the label. They are probably Type II sourdoughs - use commercial yeast and have a lower pH than a San Francisco Type 1 'sourdough'. Sourdough bread by the way is better for you than unfermented breads.
Please though tell us about your recipe. There seems to be a lot of flour for a bread-making machine for one batch. How large is your machine? Is there any way you can post a picture of a finished loaf (perhaps cut and uncut) for us to see? What kind of texture/crumb does it produce with all that butter in it? Is it somewhat sweet with that much sugar?
