For example, the English kitchen [китчен].
Is this way suitable for children?
When you use a transcription system, it should be appropriate for the language that you use it for. This means that is should at least be able to represent the sounds used in the target language. For example, the official transcription systems for Standard Chinese is hanyu pinyin. Hanyu pinyin uses the Latin alphabet, which is used by many other languages around the world, so for many learners (though by no means all learners!) it has the advantage of being familiar. However, pinyin uses the Latin alphabet in way that are counter-intuitive for many learners (e.g. speakers of Germanic or Romance languages).
A transcription system for English would need to be able to represent all the sounds used in the English language. Such a system already exists, namely the International Phonetic Alphabet or IPA. IPA provides a very accurate representation (not just for English), however, its disadvantage—especially for children—is that you need to learn an additional "alphabet". So the wish to use the Cyrillic alphabet that children are already familiar with seems attractive. However, as some commenters have pointed out, the Cyrillic alphabet cannot represent some sounds in the English language, so transcriptions based on it would be incomplete or misleading.
Instead of using the Cyrillic alphabet to represent the sounds of the English language, I see two alternatives:
Generally speaking, transcription is a good approach if the language that you want to learn is not a written language, or for languages like Arabic where the writing does not represent all of the sounds (for example, mumkin is written mmkn in arabic).
If the target language is simpler than the native language, transcription may be useful in parallel with the writing of the target language during early stages to give a guide about pronunciation.
Learning a language entirely orally works well with young children, as they learn vocabulary organically without the need for written lists. For adults, vocabulary learning is often the biggest barrier to learning so a personal vocabulary list is usually necessary, and (apart from the types of language specified above), it is best to just do some work and learn to write in the target language from the start.
Such transcription is nearly useless. I.e. "kitchen" is in IPA: "ˈkɪʧən" which is closer to "[кичн]".
Also, "beach" and "bitch" would be phonetically transcribed to "[бич]" because Russian cannot show the difference between "biːʧ" and "biʧ".
I know from personal experience, that for Russians is hard to distinguish between "bed" and "bad" because Russian language does not have the vowel "æ". So how you would transcribe it in your proposed system? And many other sounds, like "ð" vs "θ"
We have IPA for a reason.