As a counter-anecdote, I've never had a 6-15 or 6-20 in any home I've ever lived in. I'm aware of its existence and utility, and I even have a duplex Leviton 6-20R in my Amazon shopping cart that I will purchase and install in my kitchen if life ever allows me to be to be a homeowner again, but I just have never encountered them much in the wild. (Except once, over 20 years ago: The wall-mounted AC in a hotel room I stayed at in Florida used NEMA 6. [EV owners love this one simple trick!])

I have issues with typical US socket arrangements. They begin to disconnect too easily when things move around, as small appliances in a kitchen tend to do. They still work when partially-disconnected, but they have exposed live elements in this state.

This has probably happened to me hundreds of times so far. And while it hasn't harmed me yet, and I'm not particularly afraid of 125VAC, I'm always aware of that potential harm when I encounter the condition. I have also found (and corrected) the mythological problem where a metal thing has slid down a wall and onto a partially-inserted plug.

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You know, as I wrote what I wrote before and I write this now, I've reviewed many of the world's plugs. Some seem kind of OK, some are huge for no apparent reason (adding both cost and bulk for little benefit), and some seem like a series of trade-offs that are based in legacy.

And the US certainly has its own legacies, too:

Like you alluded to, multi-speed 4-wire outlets, offering hot, hot, neutral, and ground. In a fairy tale world where these can be compatible with bog-standard NEMA 5-15P appliances, that becomes a very, very easy thing to sell. It becomes so easy to sell that the ease of use starts to become more important than some of the other functions I've considered. They can even configured so that each of the 120V portions operates on a separate leg. (This ruins global compatibility, and will ultimately require revision of the NEC since split-phase duplex outlets are presently only allowed in kitchens, but boy would that be easy to sell. It also requires 4 wires instead of 3.)

Is there enough room to add a contact (or two) inside of a 5-15R, using modern manufacturing methods? Maybe a square hole near the middle somewhere with positive engagement?

> I've reviewed many of the world's plugs

https://www.plugsocketmuseum.nl/

> They begin to disconnect too easily when things move around

Replace your outlets, the old ones did this, but the newer ones, especially commercial grade (which is only $2 more), hold much better and they won't go loose.

> Is there enough room to add a contact (or two) inside of a 5-15R

I was going to have it like:

    -.
  | | |
   .-

You can plug in a 5-15 on the left, or upside down on the right. And you can plug in a new 240V outlet into all 3 vertical lines, plus the ground as a horizontal line. (I can't show it on the ASCII art but I would have the ground much closer to the powered plugs relative to the round ground pin, to make the plug smaller.)

The cord for the new 240V plug would exit from the left or right, instead of straight out or from the bottom like current cords.

This would make the plug long and narrow, and inline with the cord, to make it smaller.

I would have this outlet always be 20A, there's no reason to have this 15A stuff we have now - that's just legacy. But if we really needed to, the center slot could turn into a + shape to enable 15 vs 20.

For NEMA 5-15: Better outlets do help, but do not eliminate the problem: A tighter friction fit is still a friction fit, and given enough wiggling they still tend to work themselves loose -- just on a longer timescale.

A connector with a positive snap-action would seem better: It is either snapped tightly into place, where it is connected and it works. Or it is unsnapped, where it is electrically disconnected and free to fall out completely. USB C seems to get this part right, as an example; good versions of the C7 ("mickey mouse") power connector seen on many laptops and boomboxes of yore also seem to get at least part of it right.

Anyhow, I don't think it can happen with a new combined outlet that preserves 5-15P compabibility, so...

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Your combined outlet idea is neat, but it increases the minimum size by 50% and that increase may require new boxes, and definitely requires new coverplates.

I'm thinking something more like this arrangement:

   | o |
       
     U

Key: | is existing 5-15-style blade opening, U is existing 5-15 ground, and o is the new terminal on the other hot leg of split-phase.

The new terminal can be a blade, but a square or rectangular cross-section seems better: It's still flat for ease of making good contact, but it's also strong in more than one direction. Having a shape for that hole that is incompatible with existing blades also helps prevent Curious Little Jimmy from bodging a 5-15P into the wrong holes and torching appliances by applying twice their expected voltage.

It preserves 5-15P compatibility, does not necessarily increase the space used inside of a box, and fits behind the existing (and vast) array of cover plates. Grounding remains optional, and polarization and remains optional for non-grounded things (and both are determined by the plug end, not the outlet end, just as today).

The ground terminals always face the "outside", which is perhaps safer, and the inverted outlet

Still to sort:

Tamper resistance mechanism. The NEC requires that mechanism in many cases, these days, and universal application requires NEC compliance.

Also compatibility with existing wiring. One advantage of using a single-purpose outlet like 6-20R or even L6-20R is that existing wiring can (in some cases) be re-used by re-identifying (eg) the white wire for use as a hot leg, and this can speed adoption in some existing dwellings. But both of the layouts we've drawn have a neutral terminal on their face and invite the insertion of 120V plugs, even if they won't work.