Just like other have said, the original film is actually much higher quality than what we consider HD. Think of the actual original film roll as the master print, the highest visual quality rendition of the movie that exists anywhere.
Now its only when we try to copy this master print that the quality decreases because of technology limits. At first we copied it to VHS, then to DVD (yes I know I'm missing some stuff like laser disk but bear with me) and now we are putting it on a blu-ray disk and player. Its like taking a picture of the Sistine chapel with your old cell phone camera, then later coming back and taking a picture of it with our brand spanking new Canon DLSR camera. The quality is much better, but still not as good as seeing it in person.
As our disk and player tech continue to improve, we will continue to approach the level of resolution of the original master print. That's why for now we are at 1080, but in the future will have even higher resolutions!
The way we've defined HD today is 1080p (720 as well but let's pretend it's only 1080 for now). This means that we have a pixel grid with a vertical height of 1080. Following the hd aspect ratio standards we need to have our width follow the ratio of 16:9.
So we have now defined our tv screens to follow this standard. Imagine 2,073,600 tiny dots, arranged in a grid (1920x1080) each with a color. When we zoom out we can see a very detailed and clear image. To keep up with standards and to be "on the cutting edge", many companies switched their cameras to digital sensors. Since we can define how many pixels will e in a sensor, companies used the bare minimum to pass those standard and call them "HD" (which is not a bad thing).
Lets step back a bit. Before, we were recording movies with film. What you need to know is that film works off of emulsion. Essentially light hits the film in a certain way and devours parts of the film. These are how images are formed.
Now we are not recording how many dots there are in a grid, but how many atoms in a grid. That number is much higher than 2 million. If we were to project the same image that was taken by one film camera and one digital camera, all variables being the same, the film camera would look better.
So when they restore old movies, they scan in all of the film, do some digital touch-up work, and in fact, shrink it down to 1080p. So the process is actually trivial and does not require the entire film to be redone.
Part of this is that when you downsize the image to standard "HD" quality it looks much better when you start with a huge image. The other part is future proofing. The fewer times you need to re-scan the same film for better resolutions the lower the cost of switching to new formats.
It's like taking a huge picture from a really good digital camera and then viewing it on the tiny little LCD that they have. It can look amazing on that 3" screen, but when you look at it on a higher resolution and larger screen it looks terrible.