No modern electronics should impart a "signature sound". Anything other than the exact signal from the source being increased (amplified) is, by definition distortion!
Amplification is a mature science and all of the major players will provide a clean neutral sound as the foundation.
Now once you start involving YPAO or Audyssey and various processing like DTS or Dolby there will/can be differences (I don't know how strictly Dolby and DTS control implementation of their products). However, if you run the AVRs in Pure Direct mode and they all have the power required to properly drive the load presented by the speakers, they should seriously sound identical if all other variables (such as speaker location and volume level) are controlled.
However the 2060 has established itself as a fine AVR and there is no reason to believe the 2070 would not be as good.
Personally, I would go for a refurb Flagship A3060 for $1200 with a 18 month Yamaha warranty. Note that these are reconditioned by the manufacturer (Yamaha in this case). Third party refurbs are pretty sketchy, but I have bought over a dozen manufacturer reconditioned units from A4L and they have all been great!
https://www.accessories4less.com/make-a-store/item/yamrxa3060bl/yamaha-rx-a3060-9.2-ch-x-150-watts-networking-a/v-receiver/1.html
It is up to you. I just want to make sure you considered this option. Most of us who have been doing this for a while realize that it is rare that the current model (A2070) offers anything important over last years model (A2060) while there typically is a worthwhile difference in going up to a better model (2060 to 3060).
You can also still buy a new 2060 for $1300 (Amazon). I assume it is discounted as an obsolete model.
Good luck, but none of these options suck. You are choosing between various degrees of good!