What are the virtues of getting a 1080 Ti Desktop over buying a PS4 ?

Ok, a few misconceptions here.

1) A Graphics card is a big part of a gaming machine, but it's only one part. It doesn't matter to keep upgrading forward if the interface (mainboard) and the cpu are causing issues or bottlenecks in conjunctions with the gpu. Not to mention cabinet air pressure and airflow and memory type, capacity and channels. A PS4 is built for gaming, a computer can be built for home, office, production and gaming.

2) The people who said to you that you need a 1080 Ti or a 2080 Ti for gaming is either lying or are ignorant. You definitely don't need either. I have a 1070 that I bought for $350 2 years ago and I am running games at ultra 1080p just fine, even at 1440p. And this is at greater framerates above 60, something a stock PS4 can barely achieve in a handful of games due to it running a custom AMD Jaguar processor from 2013, so does the PS4 Pro, and the Xbox One X for that matter. It leans into what I mention in my first statement: A GPU is only one part of a system for gaming. Consoles also make heavily use of a whole spectrum of technologies to make games run on them, like checkerbox-rendering, dynamic resolutions and shader points. They also don't run at high quality settings compared to what a PC can achieve on ultra. Usually they are hovering at the medium range, with a set to higher textures because they have good memory buffer.

3. The original PS4 has 1.8 floating-points calculations in compute performance, similar to a Radeon HD 7850, a card from 2012. In comparison the 1080 ti have 11.3. The RTX 2080 Ti have 13.4. Of course we can't take the teraflops and have an exact performance measurement in gaming since consoles are built as a single unit, but the point I am making is that a console can achieve great performance levels on low to average hardware because it's optimized for it. A PC can have thousands of different configurations developers need to take into account. Optimization is key to make use of the available hardware. With that said, looking at this statistically, an original PS4 doesn't even come close to the raw performance of a 1080 Ti or a 2080 Ti.

I agree with @Faith -

If you could get yourself a great deal on a GTX 1080 (not the Ti - which is a bit buggy in my opinion) that's what I would choose.
GTX 1070 is also a great card. One important item is the spec on memory bandwidth in bits. You want your card to match the Mobo in this aspect.

Cheers!

What I am neglecting? The 1070 isn't even equal to a PS4, a Radeon HD 7850 is, or a RX 460, or a GTX 660. You can get or build a whole PC to the same price as a PS4 with equal theoretic performance (there's plenty of people who have done that over the years), but since the PC is a configuration compared to a PS4 that's the configuration the PS4 have a greater advantage to optimization because it's easier to develop games on a single unit. Console manufacturers also have better deals on the parts they are using, reducing costs. Sometimes they may sell consoles in loss so that they can get more users on their services and platforms instead. Price is a bit of an issue when comparing PCs to consoles, it's a bit muddy, but they are definitely not three times more expensive for the same performance as consoles. If you are building a system that is similar to a PS4 in price and performance, you could back then and you can today.

PC = Computer
Game console = Toy.

I have gamed with a PS4, my brothers. I have gamed with a 1070 until a few months ago moved up to a 1080Ti. The current 1080Ti games far better, and is worth the price to me.

If a PS4 is a good experience for you, then nothing is wrong with that. Seeing how this a forum with more computer users here you can guess what most prefer.