@Minty
As to definition:
There is a distinction between assault weapons and semi-automatic weapons. A semi-automatic weapon requires one trigger actuation for exactly and only one round to be fired. Neither type of weapon generally requires a manual operation to reload after one round has been fired.
However, if you look it up, you will find - for most legal definitions - that "assault rifles" have the ability to "select fire" which implies full-automatic operation, the ability to fire more than one round on a single actuation of the trigger. So, for example, if you see headlines about the USA's popular AR-15 rifle, that is a semi-automatic weapon but not an assault weapon because it does not have a full-auto mode or short-burst mode. With a rifle, most purists don't care that it looks impressive. They care how well it shoots.
The news you hear makes you ask why someone wants "to legally possess a fully automatic assault rifle." But we don't. Full-auto weapons require special, more stringent licenses. The shooter incidents you hear about are committed by folks with semi-auto weapons.
As to why someone would have a semi-auto weapon:
In target shooting sports, your target isn't usually going anywhere if you miss the shot. Though there are competitions with moving targets, there are also a lot of contests with stationary targets. If you take a fraction of a second longer to ready the next round, it is not usually going to result in a terrible score. Accuracy usually counts more. Therefore, semi-auto weapons DO show up in competitions but other types show up as well.
But what about other sporting uses? When you go deer hunting as my brother-in-law frequently does, if your first shot misses, your second shot is quickly ready whereas a bolt or lever action rifle takes longer, makes extra noise, and gives your target more of a chance to get away.
If you hunt feral hogs (commonly done in Texas and Louisiana and Mississippi), the ability to shoot quickly means you can more easily get rid of those hogs because their herds scatter on the first shot. Feral hog hunting is legal because they are classed as an invasive nuisance animal that eats whole fields of farmer's crops.
If you go hunting for nutria, you had better hit with the first shot or have a semi-auto weapon handy, because they will attack you since they are territorial. Again, nutria hunting is legal because they are invasive nuisance animals that destroy natural vegetation and thus contribute to land loss in the marshes by eating plants that normally would prevent erosion. It doesn't help population control efforts that they are quite fertile little buggers.
While hunting alligators, they will go into a frenzy if your first shot isn't a kill shot. Their natural ridged armor protects them and makes the kill shot harder to assure. On the show
Swamp People, they sometimes show that a gator's natural scaly armor deflects the bullet or shatters it and the ricochet can wound the hunters. So you had better be ready to take another shot real fast. Once upon a time, the American Alligator was a protected species, but they rebounded so well that now there is actually a population glut of them and Louisiana has a yearly month-long hunting season for them, hoping (but lately, not succeeding) to thin out the population.
In all of the cases I named above, I have only chosen hunting of animals within a day's drive of my home, a region including the western Gulf of Mexico coastal area. Now multiply that by how many different hunting regions there are in the USA. You will find an equal number of similar reasons for people to legitimately have semi-auto weapons.