I've been looking for new shoes again and the one thing that always struck me about cycling shoes is how uncomfortable they are. I started to wonder why nobody makes a cycling shoe like a running shoe, and I'm not talking about a shoe to walk in, but a shoe with enough padding that you don't feel every crease and bump, with a padded tongue, sorbothane type insert, and still with a stiff CF sole. Cycling shoes are still in the dark ages compared to running shoes. I don't want my foot to be loose in them but they don't need to be so tight as to be uncomfortable. And again, these aren't MTB shoes, or recessed cleat shoes for walking I'm talking about, but real road shoes that have internal padding and contours for your feet.

Is there something out there that I haven't seen because they are over my usual budget?

I guess I titled this Shoe Inserts, but either inserts or whole shoe would be great.

The manufacturers have gotten quite clever. There are many examples of what you are suggesting. Specialized Body Geometry is just one. You buy the road shoe and it comes with a neutral (completely flat) foot bed insert. You can then buy a low, medium or high arch support insert to swap. Or go one step further at selected Specialized dealers and get a custom thermo-molded insert. All the other major brands have similar options. Or just buy a third party insert from a shoe store or athletic store and experiment. Many, many options.

As far as overall comfort; good socks, practice fitting the correct shoe and setting up the cinch just right has made shoe comfort issues a thing of the past for me. Reference -- I'm using Specialized Torch 2.0 and a Body Geometry Red insert.

I have knit running shoes. Love those things, they're incredibly comfortable, and great in the water too.