The Wii U might be able to have DVD and CD playback through the Wii Menu if you have used homebrew on it. Haven't tested it yet though

I say yes.

One of the shortcomings of the Wii U is that it doesn't have DVD and Blu-ray playback. I'm not sure about CD playback, but who would care if it did? The disc drive on the Wii U really doesn't do much beyond play Wii U and Wii games (more on that in just a sec). But there's more to it than that, it ultimately adds to the manufacturing cost and means lost revenues because of costs of distribution for the games being sold (returns, cost of goods sold etc.).

I believe it's possible for Nintendo to offer a digital-only WiiU with a bundled game at $200 (or even less) that would fare far better in the marketplace  than the current WiiU SKUs. The advantages of digital-only would allow them to price the console and games more aggressively, while maintaining the same bottom line for each console and game sold.

Q: What would retailers think? Wouldn't they be mad they can no longer take a cut of the game sales because of digital distribution?

A: I don't think they'd really care. If they have a product that has more consistent inventory turns than the current Wii U and its slate of games, they'd actually be happier with a digital-only Wii U. You could always still sell download codes for the most popular Wii U games (Mario, Zelda) or package them in for games like Skylanders.

Q: Wouldn't that ultimately mean less shelf space made available for the Wii U? Isn't that bad?

A: It would likely mean less shelf space, but I think retailers could re-allocate some of that space to where it counts the most: offering a playable Wii U in store. It's far more meaningful for people to get hands on with a game like Mario Kart 8 then to have a copy of Splinter Cell Blacklist U staring the customer in the face.

Q: Wouldn't this possibly hurt third party sales? I mean if they don't see that the game is available, they won't buy it.

A: Few people buy Wii U third party games. That's a fact that isn't likely to change. For the few 3rd party sales made for the Wii U anyway, they'd also be better off with digital distribution as the primary sales channel because they too could earn higher revenues with digital only. If the retailer desires a physical retail presence (like say Ubisoft with Just Dance), they can retail download codes directly. But for now, 3rd parties and retailers look at physical WiiU games as liabilities -- inventory that won't sell unless a fire sale is started - that's not advantageous financially. However, since download codes aren't necessarily inventoried, it makes the financial statements look more favorable while providing the same choices to consumers.

Q: Wouldn't people miss the disc drive? The backwards compatibility? Owning a physical copy?

A: The only folks who'd care about backwards compatibility are those who don't own a Wii. The Wii's back catalog also arguably competes unfairly with new games coming into the system. For roughly a similar price, why would I buy Batman Arkham Origins on Wii U (a fairly average game) when I could pick up a used copy of Metroid Prime Trilogy for the Wii? Nintendo not only doesn't collect a royalty for a new game sale, but they in turn receive nothing for a resale of one of their popular games. A digital only console (as well as a bigger eShop selection that might include the Metroid Prime Trilogy) could solve that problem. If people want physical copies so badly, they can buy a WiiU with a disc drive and use it. And if they want to play a Wii game that wouldn't ever be availabe on eShop, they can buy a Wii.  That simple.

What are your thoughts?

Retailers are not going to promote a digital-only console, unless they receive a hefty cut of the selling price. This means that it's more likely that the Wii U's price will go up, rather than down. Or Nintendo takes significantly larger losses on the hardware which completely offsets any gains from higher digital software sales.

With a digital-only console, retailers are not only losing out on sales of new software, but also on sales of used software. Your mistake is that you believe that retailers would welcome digital-only, but the very opposite is the reality.

What would need to change with the account system exactly?

Standalone Blu-ray drives similar to the Wii U's take roughtly about  $30 -$50 to manufacture if you do some back-of-envelope calculations.

If you remove that component from the finished product, it makes the Wii U cheaper to manufacture since there are fewer parts to hold inventory, fewer QA tests needed, and less labor to put the whole thing together.

You missed my point on download codes sold at retail. Just provide the customer with a gift card thingy and get the download code upon transaction taking place.

Retailers don't gripe at the fact they can't sell mobile and tablet apps directly. Instead, they focus the retail experience on hands-on trials. It works pretty well -- however because of the amount of space the current WiiU retail presence takes, they can't cede more floor space to this.

Selling a digital-only WiiU would be not much different from selling a Kindle Fire in terms of the bottom line per unit.

The gain from removing the disc drive would be overwritten by the need to increase the hard drive space (in fact, cost may actually increase)...You are proposing this idea for 200$ but that would just make Nintendo lose a ridiculous amount on each console.

All the questions in the OP are also valid concerns that make me think "why would this be a good idea?". You listed a few reasons (or rebuttals) as to why it isn't a bad idea but why is it a good idea? I can't think of anything

EDIT: I see you are talking about increasing gain from software sales...but that is sort of silly. The people who would buy this model are the same people who would buy digital anyways on a normal Wii U...its not like the Wii U doesn't offer digital sales of retail games already