Here is my honest opinion as I recently upgraded from a 3600 to a 5800x3D myself.
It is 100% worth it. Now the performance gains you see are going to vary from game to game. Some games you will see a noticeable improvement to FPS and others a smaller improvement like 15-20 extra frames. The main difference you are going to notice is the 1% lows. This chip absolutely is worth it for that alone as everything is so and I mean SO much smoother. Less frames drops (almost nonexistent) and now I feel like my GPU is finally being used at its full potential (5700XT which I was also told paired well with a 3600). That 3D cache in this chip makes all the difference. So yes, completely worth it.
Just make sure your BIOS is updated to the latest version to support it if you have an older motherboard and be sure to buy a good cooler, preferably an AIO as this is a spicy chip and the temps can get up there. I have the Lian Li Galahad 240 as my cooler for it and in game the temps range from 50c to 70c depending on the game.
TL:DR
Do it! Best purchase you can make and it is definitely still future proof for many years to come.