Yes, the image is twice as big...if the magnification is true x2. More expensive scopes will be more true to power. Check my math, if you like trigonometry, but here it is.
A 1m target at 100m will have an apparent angle of 0.573 degrees.
A 1m target at 50m will have an apparent angle of 1.146 degrees.
So when the target appears 1/2 the distance, the angle to the target is twice as big. But when we think about objects doubling in size, we really expect it to double in all dimensions. That would be 4 times larger (your target is 2 dimensional in your scope focal plane).
The answer is yes, x2 power optics are not very powerful. A target at 50m is still pretty small with a red dot aim point. The usefulness of a red dot at distance also depends on the size of the dot. Some people prefer small MOA dots so they can hit farther targets more easily. Some people like larger MOA dots so they an see it more easily, but shoot shorter distances. An experienced rifle shooter can learn to center a large sight on a small target. When I shot 600m (iron sights), the front sight post was about 3 times wider than the bullseye.
Red dots are easy to pick up when changing targets rapidly. They are for close work. If you want to see target detail at 100 yards, start at a x1-4 scope with a reticle designed for your intended use. x6 should be good to 200 yards. Most people like to over magnify.