What is the point of using interceptors to attack UFOs?

Solution 1:

It is in fact an event where you will get a chance to get more resources. This is really important; you need to acquire certain materials to do certain research to get a path to victory in the game, and that says nothing of getting your soldiers properly outfitted.

As to the randomness, that's not likely. The aliens have an agenda and it will play out on the geoscape. UFOs have missions; shooting them down prevents them from carrying those missions out. Allowing the aliens to carry out those missions will help spread panic/alien presence (at a high level) which is bad for you.

I have only ever seen ufos where I had satellites deployed. Eventually you will need to upgrade interceptors as they will start encountering much larger, more powerful craft.

Solution 2:

Just to add to some of the answers here, missions where you shoot down UFOs are excellent training missions for your green inexperienced soldiers to get some levels.

Solution 3:

It's very important to keep on top of the incoming UFO's, else they will get increasingly bigger and shoot down your satellites (as well as increasing panic, etc.). At this point you're on the path to losing, which in XCOM is frequently non-reversible.

It's frustrating, because if you get stuck at a point where you didn't pre-invest in the right resources to shoot down UFO's, no amount of strategy will help you: you just lose. Unlike the combat-based aspect of the game, where good strategy (and luck) can overcome bad odds, the base management portion requires extensive "psychic" knowledge of the aliens' attack timing to plan and build accordingly, else you're just screwed.