On small prints you are dealing with macro tonal relationships. As the print size increases you start to deal with micro tonal relationships. (This is in relation to the negative, not the print.) Remember, you are enlarging the space between the grain. In a small print the space is imperceptible. As the print gets larger the space between the grain starts to overwhelm the grain. This is why you need to print "lighter and with more contrast".
To understand this, make a small print then elevate your enlarger all the way up and print a detail of the image. Compare the structure of the two and you should see what I am talking about.
I think one problem you are having is that you want the larger print to match perfectly to the smaller print, but it will never happen. If you want to make big prints, then just make the big prints. No one except you will ever hold the small print next to the big print!
The other thing I would suggest is that you evaluate prints at viewing distances. Holding a 20" print in your hand will disappoint you from the start if you are not used to it..
Hope that helps. I hope it makes sense too. It is late as I write this.