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Personally, I teach orbital mechanics classes to preschoolers, elementary and middle school kids using a makeshift trampoline with stretchable cloth clamped to the rim. Place a heavy weight (e.g. a dumbell) in the middle to simulate a large massive body like the earth or the sun. Use marbles to illustrate a spacecraft or planets. You can easily show the basic relation and variation in velocity as a function of distance in an orbit trajectory by rolling the marbles in elliptical, hyperbolic or (if you practice really hard) a parabolic trajectory. Giving the marbles a push while in “orbit” also shows how impulse thrusts affect the shape of the orbit too. More importantly, this demo helps give kids an intuition into einsteinian physics, where the presence of matter bends the space around it and is the reason why objects naturally follow curved paths in space instead of straight lines.

See this YouTube video for other ideas you can illustrate using this kind of demo

They can be easily constructed using PVC pipes and stretchable cloth fabrics. See these instructions for constructing your own gravity well. In my experience, the larger the well, the easier it is to demonstrate orbital mechanics.

UC Boulder has a project, PhET, that has many free, interactive, in-browser math and science modules. They have one called Gravity and Orbits that's written in HTML5, making it compatible with most modern browsers (including Safari on iPads). If you click the 'For Teachers' drop-down, you'll find it even has quite a few resources for lesson planning with it for middle schoolers.

I think this will suite your needs of it being appropriate for a class room and browser-friendly (and free!).

As a side note, if you want to tie Gravity and Orbits, or other software you end up choosing, to how the entire Solar System works, NASA has a great interactive site called Solar System. If you click the concentric circles on the menu bar, you can see the planets (and more) orbiting our Sun. You can also zoom in and out like if it was Google Maps. You can click a planet (or other object) to find out more information on it.

Spaceflight Simulator

An Android software, but it's possible to run it on a PC as well.

It's 2d, so it's much simpler to use than 3d software.
There are some premium features ($4 unlocks all of them forever), but the free version is enough to launch missions to all the planets in the inner Solar System, and to put space stations in orbit and dock to them.

The learning curve for simple missions is very easy, the number of basic parts is very small. You can achieve orbit within a few minutes of having installed it.

It's much more simple than the Kerbal Space Program, but for the purposes presented in the question, that seems to be a big advantage.

Probably too basic for what you want but Osmos (https://osmos-game.com/) has modes where the player is basically orbitting an attractor and has to change orbit to rendezvous with other objects in different orbits. This teaches the basic idea that you can't just thrust towards your target and expect to get there. Another mode involves multiple attractors orbitting a central one and the player has to set up a transfer orbit between them.

As I said this is probably too basic and is probably not very accurate but it may serve as an introduction to the idea that orbital mechanics work very differently to how you move around on the surface of a planet or even in the air. You might find it useful to let them play with this to learn that guessing isn't a great way to move in space and that a mathematical approach is not only better but basically essential.

There is a free demo but I don't know if that includes the modes which feature orbital mechanics type gameplay.

I think literally the first game ever created by Stephen R. Russell (I heard about it through a book on J. C. R. Licklider) was a game called SpaceWar.

The idea is to shoot your opponent whilst avoiding getting sucked falling into the gravitational void in the centre.

There's numerous emulators for it - for example https://www.masswerk.at/spacewar/

I can't believe no one's mentioned Orbiter 2010! Its a free game, similar to KSP, but without the building part, and with full-scale, n-body physics. There is a mod for it called Interplanetary which allows you to input your location, your destination, and the maximum time you have to get there, and it will tell you how much dV you need and give you a trajectory. It has a steep learning curve however.

Here are two more fun ones which have not been mentioned before

• Super Planet Crash : Browser based game where you add planets to a solar system.The aim of the game is to score as high as possible. Score points for bigger planets (which upset other orbits more), for more planets (which lead to more complex dynamics) and for making your solar system last longer (game ends when a planet is ejected from the solar system).
  http://www.stefanom.org/spc/




• Chaotic Planets : Not really a game but more of a demo. Creates random systems of planets so you can see the dynamics play out.
  http://labs.minutelabs.io/Chaotic-Planets/

While it does not seem to offer propulsive orbit changes, the TestTubeGames Gravity Simulator demonstrates orbital motion in a web interface.

In theory you could use KSpaceDuel https://www.kde.org/applications/games/kspaceduel/ for this purpose.

The power-potential of the spacecraft is way out of range for a typical rocket, but orbital transfers work the way they should until touching a screen edge.