i just replaced r5 3600 with a r5 5600x and its only$299 and well worth it so if thats in your budget its well worth it.
heres a couple videos with my set up so with the 3060ti im sure the r5 5600x will perform very well.
 
(18) Call of Duty Modern Warfare 2019 | Shot with GeForce - YouTube
 
some cold war
 
(18) Call of Duty Black Ops Cold War | Shot with GeForce - YouTube

i use geforce experience on performance settings and it still does great on max settings just less fps. the 5600x comes with cooler but im using the nzxt 360mm rgb liquid cooler. x73
 
003 - YouTube

nzxt came red and black i have the front off looking at temps with and without it because their fronts are solid. you check them out at nzxt.com

I started off with the 5600x in my return to PC gaming and was very happy with the performance and the thermals.  It does leave a little to be desired, but if you are running a 3060 it will match up very well to that.  The only reason why I moved on from it was because I upgraded my GPU and had intentions of gaming at 1440 instead of 1080 as I had originally planned my system around.  Thermals are great on the 5600 and it does come with a decent stock air radiator as well.  If you can spare the money, a good AIO like the artic freeze or the Lian Li Gallahad would be great options if the funding is available.  If not, just make sure your case has ample air flow and you should be good. 
 
One word of advice would be to stay away from the 5800 as it runs very hot compared to the 5600 with just a small performance gain.  The 5800 was a purchase I regret in hindsight especially now since I recently was lucky to get a 5950 at 799.  I could have saved that 450 and used it for other things had I been patient.  Hindsight is 20/20, but just wanted to share my experience.