You need the 4K stream to get it to look closer to 1080p locally. Real time image compression isn't as good as encoding a movie or tv series with all the time in the world to optimize the image quality. Plus even 4K has to squeeze into 35 mbps, while 10 bit 1080p60 is a 3,560 mbps data stream. And of course the first step is downscaling to 4:2:0, which means quarter color resolution before compression starts. (Means 4K effectively streams at 1080p color resolution)
Can you select 4K streaming to play on a 1080p screen? I always see very clear benefits from running you tube in 4K on my 1080p screen, instead of watching the 1080p stream.