A great question! I love how UX design makes you think of these things.

However, just to play devil's advocate, I can list several reasons off the top of my head why we should **not** have all doors as you described:

 - **Hinges:** would have be to become more complex. One-way hinges, as they exist now, are a quite simple 3-piece design that require very little maintenance over their lifespan. As we all know, the more pieces you add to the puzzle, the more likely it becomes that one will break.
  - **Hanging:** There's definitely an art to it. There's some pretty tight tolerances required to get the door to seal properly on all sides with materials that are not analogous to having pretty tight tolerances (like wood and wood screws).

     Of course, it must still swing freely and close without obstruction. And it must open all the way smoothly without hitting the floor or ceiling. I could only imagine a door that swings both ways being double the headache to get fitted.
 - **Retrofitting existing doors:** This deserves an analogy to the USA and the Metric system. Why don't we switch? The Metric system is easier to teach, easier to learn, and easier to use.  
   
 But, all of our roads and street signs are already set up for miles! Ok, well we'd have to keep using miles. And all of our thermostats use Fahrenheit! Alright, we'll keep using Fahrenheit, but only for thermostats and weather reports. But tools, parts, and screw sizes, that'll all be metric. Wait, we still have to deal with *EXISTING* machinery and hardware stuck in these archaic sizes!

 It seems pretty impossible for a whole country to just switch, *even if it's better*. There's a big argument for "backwards compatibility" in there, somewhere.

 - **Security:** a normal one-way door has a frame that it sits in, with edges that hold 3 to 4 of its sides in place. A two-way door would have to forgo all of these edges, leaving the only things holding the door in place its two-way hinges and the deadbolt. And without edges on the doorframe, the deadbolt would be exposed.

 - **The Elements:** here we see the edges of our standard doorframe come into play again. Turns out they do more than just security, we normally put weather-stripping on them to keep out roaches and relatives and keep in things like expensive Air Conditioning and girlfriends.

But of course, I'm being silly. Obviously *outside-facing* doors will have to remain one-way. You're talking about **inside** of a building, are you not? Well, let me have another go:

 - **Privacy:** Whether it be inside a residential home for a bedroom door, or for a private office in a commercial structure, a **closed door** has universal significance. Although it may not yet be understood, us humans have a need and a right for personal privacy. The thing that assures us we have it is a latch that keeps the door shut and a handle that must be operated to open it.

So we cannot retrofit doors that face the outside world, or doors to private spaces (from bedrooms to bathrooms, I think that includes almost all the doors inside a residential building. You could argue that a residential building is in total a collection of private spaces). 

And no to personal offices. So really, our only candidates are public-to-public spaces that see a high flow of traffic. In which case I would argue, why even have a door there in the first place? Unless it's required, like an "Employees Only" section in a restaurant or a "Surgeons Only" area in a hospital. And I think those places already make use of a no-handle two-way door.

A great idea tho! Just my two cents.

>**Edit:** I **would** love to see this done on every public restroom everywhere. I always save the paper towel I used to dry my hands (if they have them) to open the door. But if it's an *outward-swinging* or *two-way* door, you can use your elbow, shoulder, or foot to open the door and keep your fresh washed hands clean.