Russian with English sounds is practically incomprehensible, and you have to understand native speakers as well. Just think of a Russian saying vafe instead of wave, debt instead of dad, вёрт instead of word, will he be always understood?
No one is suggesting speaking "Russian with English sounds", merely that the distinction between hard and soft consonants is rarely (if ever) a real-world impediment to understanding on its own, especially for beginners who are struggling with so much more already. I was using Russian for years before I really understood the difference between hard and soft consonants, let alone became able to pronounce them myself. In all that time this inability hindered me not one iota. What impeded communication was gaps in my vocabulary and grammar and mistakes with declensions and conjugations. There are literally no real-world situations where pronouncing ся as sa or s-ya will confuse a listener.

These books do not help, they only make harm.
Don't be silly. If communication is your motivation, then of course they help. If your motivation is learning the language for academic purposes, or to perfect it, or to pass as a native, or even to avoiding offending the delicate sensibilities of native speakers who are overly precious about their language, then you might have to pay more attention to it.

you have seen yourself: Боб Уайтман's table helped a learner more than many textbooks, and Bob managed to write it even not being a professional Russian as a second language teacher, as far as I understand.
No it doesn't. Боб Уайтман's table only explains the orthography of soft and hard vowels and how they relate to preceding consonants. It doesn't explain the actual distinction between soft and hard consonants at all. To learn that you have to hear it, and to be able to hear it you need an understanding of the basics.

It is for a learner to decide what is important for him and what is not, textbooks must say correct things. Why should be the pronunciation fully negected and the grammar be studied if Russian is a living language?
Who has said that pronunciation should be "fully neglected", or even anything like that? You asked why beginner texts tend to ignore the subtleties of soft and hard sounds, and I attempted to explain it for you. It's not about neglecting anything, it's about prioritising that which makes a big difference over that which makes less of a difference. A beginner text can only contain so much information. Of course an author could include an in-depth explanation on softness (not that it's really possible to explain the difference between a soft and hard sound using only text anyway), but they'd have to lose something else to make space for it. What would you suggest should make way for this explanation? The chapter on verbal adverbs? Numbers? Reflexive verbs?