Bigger rims (bigger diameter and width) do provide advantage in terms of handling.
-With bigger diameter, you'll need to use lower profile tires which has less flexing/stiffer sidewall than normal thick tires. This will give you more direct/sharper response, better grip during cornering, and increase stability/reduce the feeling of floating.
- Then the wider rims surely also need to pair with wider tires which of course gives you more footprint, so again improves grip even further.
- Wider & lower profile tires also are normally higher performance rated with better compound and strength for high speed, hence again you improve the grip and handling of the car. Of course there are still some bigger tires which are just big without really having better performance than smaller tires hence negates this advantage. For example, if you compare 17 tires from those cheap China brand tires with 15' Michelin, Yokohama, BS, GY, etc., then of course the 15' will still give much better grip.
- Finally some cars do really need bigger wheels because they have big power and need big brakes. Small rims will not be able to accommodate those monster size brakes, hence the big rims is mandatory.

Note though going too big indeed provides disadvantages if it ends up being way too heavy and tires too thin. The extra weight will reduce acceleration, worsen fuel consumption, increase braking distance, and actually might worsen handling due to more unsprung weight which the suspension has to control. If the tires are too thin like wafers then of course it sacrifices a lot of comfort and will hurt your back and also endanger your rims when hitting potholes, hence what's the point? That's why those big rims which are forged and very lightweight are expensive. An example is the set I'm using now. It's a 17x8.5 forged Japanese rims but only weight 7.5kg, compared to the standard 16x7 which weight 10kg! So despite going bigger and much wider, I still managed to save a few kgs and and improves my handling, stability, and cornering capability so much more than before.
So if want to go bigger, try to use the lightest rims you can afford and don't go excessively big (generally max +2 inch in diameter). For some cars, they already use big rims with quite low profile tires so that general rule does not apply and do not upsize the diameter anymore. Though you may increase the width if you want/necessary.

It's like a nonstop trend that people keep going for bigger rims! Actually 17' is becoming rather conventional & not so big anymore now since many std cars already come with 17 or even bigger! Again it depends on your car size and how thick the tires should be to accommodate the rims. My car can still use >11cm tire sidewall even when using 17' without increasing the overall rolling diameter, hence the comfort is still good. Many std cars that come with 17 still able to be combined with rather thick tires too so that's fine. However many people just try to follow this trend no matter what car they drive, hence often become painful and actually worsen the car instead of improving it.

Btw no matter when I was using 15 or 16 or 17, I always use my tires until only about 20% left which is usually at the 4th-5th year of usage! Changing them while still at 50% is just still too much waste IMHO as they're still very good.

Agree! Small cars using big rims also IMHO looks out of proportion already!

F1 & Indycars use small diameter rims/thick tires as part of their suspension movements as well to absorb the bumps cuz their suspension is already ultra stiff.