I'm gonna be controversial and say maybe or yes. I think using a cube with poor corner cutting can definitely help to improve turn accuracy over a period of time. Try solving on a guhong, you'll lock up like crazy. Then go watch Feliks do a 7.23 ao12 on a guhong and check out the insane turn accuracy.
Obviously I'm just speculating, but those are my observations.
How could I disagree with a top speedcuber? But I am going to... just sort of.
I agree that training with worse cubes (Guhong, even a Rubik's brand) can help you be more accurate but, will that be helpful to be faster with a better cube? Only slightly, I'd say. In the end your turning style has to adapt to the cube you're using or else Feliks could not be faster with a Gans 356 than with the Guhong, if he's just turning with the same level of accuracy and not taking advantage of the improved corner cutting and flexibility of the Gans.
So, will it be more difficult for somebody that has never used other cube than a Yuexiao or Weilong GTS to get good times with a Guhong, sure, but I think he/she would get used to it and would adapt their turning style. I mean, it's not like you will spoil your turning style and will never be able to fix it. So, in summary, I agree with you but coming back to the title of the thread, I disagree that there will be any permanent damage.
I also agree that big cube practice can help with accuracy but it's a completely different muscle memory because of the different size of the cube and the outer layers so maybe the help comes the other way around, a Guhong can help you be more accurate from the beginning with a 4x4 but maybe not the other way around? Just guessing here.
I am a patzer with either a good or a bad cube but I have used a YJ Chilong as main for more than a year. Probably not as difficult as a Guhong but ages away of latest cubes, for sure. I also enjoy solving at times an old Rubik's brand with 0 corner cutting and I'm still trying to get sub-20 with it. But I do this for fun, not because I think it will significantly help my turning style.
By the way, I hate how most modern cubes with squared-off corners catch at times, by the way, if you are not careful e.g. when right thumb pushing F' like at the end of T or Y perms.
EDIT: Drew, you just made me want to buy a Guhong. Are any other old Dayans worth a try? Not as speedcubing main, just as a collectible but one that's usable as well, not just to show it on a shelf. I mean the LunHui and others (I already have 3 Zhanchis).