The basic reason of not meating meat is non voilence. Hinduism preaches non violance of any form. be it physical or mental. In my whole life i never heard of the reason you stated above.
Also i have never seen any hindu not driking cow’s milk.
I was a veggie until last year just because of my own principles of non voilance now when i eat meat i dont discriminate. cow, pig, fish, birds all are same for me.