You'd think that these publishers could more than handle what you and I would like to publish, but they cannot. Besides being highly selective (look at the authors listed below), they have limited resources, regardless of how big they are and how many names are on their door.
In the last couple of decades, smaller publishers were gobbled up by conglomerates, that weren't neccessarily in the publishing business, and the Big Six came together. What often happens when big guys consume little guys, is a certain amount of homogenizing when the big boss has in his mind what his company is supposed to represent.
While we'd love to have those huge names in our portfolio, it's difficult to reach that high until you've proven yourself in a big way. So know the big publishers and understand who they are and how huge a ground they cover. Then take a look at mid-sized and small publishers.
Smaller publishers are open to more diversified voices, lesser known names and risk. You can reach the editors with a phone call, discuss marketing, and keep your book in print a lot longer as you fight to create that platform.
These days, it's fast becoming a career path to self-publish, jump to a small publisher, then become recognized by a large publisher. But you can choose your own path. Sell a gazillion ebooks and jump to St Martin. Publish with Pocket Books and drop down to self-publishing once you have your platform. But whichever turn you make, whatever path you take, understand all the players and levels. Informed decisions are always the best.