K&N Performance Filters

As stated, cooler and denser air supporting more engine power is not a function of the filter, but rather how the air is routed into the engine (and through the filter). K&N does offer kits for many cars for rerouting the air intake to a cooler location. The K&N has better air flow as the result of a design that has more surface area. It is inherent in that design that it can effectively handle more dirt before clogging, versus a standard paper filter.
The materials are also more robust than pleated paper providing longer life.
From a pure cost basis, and changing out a paper filter every 12,000 miles (personally, I would not run one longer) the K&N probably has an edge.
But it's not that great. I use synthetic oil too, but the the advantage over premium conventional oil for MOST climates, is probably negligible. It all gets down to personal choice.

The only real advantage of the K&N is that it's efficiency will remain higher as the contamination load increases. I am not convinced it is as efficient as a well designed wetted-foam filter.
Either would be superior to a paper filter in this regard, the HP hype notwithstanding.

Naturally such after-market filters lend themselves to intake tuning much more readily than the factory setup. How much an improvement, is questionable IMHO. But from a pure design concept there should be some, whether it is significant or even noticeable can only be borne out with actual measurements for comparisons.
All such improvements can be additive. The biggest changes probably from exhaust or whatever else can be done to the engine without compromising emssion or dirivability requirements.
Therin lies the rub.