The answer, quite frankly, is no. They do not. Particularly when you're used to larger mouthpieces, switching to a small one can (at least in the short term) worsen both extremes. You could theoretically adjust to this, though you'd be dealing with a smaller, less vibrant sound than on a larger mouthpiece.
As for brands: yes, it is somewhat 'easier' to play high on the Reeves 43C and Yamaha Shew. They slot a bit easier than a Bach or Blessing up in the stratosphere. However, there are minor differences.