Written English is not a Phonetic language; you need to understand that before you go any further. As a result, while phonics are a useful aid to beginners in reading/comprehension, they will always be imperfect just like any shortcut.
If you look at the etymological breakdown of English, it contains words which we can mostly trace back about equally to either a Latin, Greek or Germanic root (there are a few others as well, but those three account for over 75% of English words.) Because Written English is not an alphabetised phonetic language each imported word retains many of the spelling rules from it's parent language.
Take the two words you've compared.
Elbow is an Old-English word and is rooted in the 'alinobogan' language which underpins Dutch and German but is neither.
Cow is German rooted, but only came to Britain after a great deal of change in the German language and has different spelling rules as a result.
Differences like this which occur all over the language make it
impossible for any rule to be universal unless we threw out all our
spelling rules and used phonetic English. That isn't going to happen because Spoken English
is wildly divergent; consider Glaswegian or Cockney. If they were to write the way they speak communication would become very difficult.
Essentially 'synthetic phonics' is thought to be the best method currently available for teaching English and claims give a correct or 'good enough' pronunciation of around 85% of English words, plus it is relatively quick. There is also another phonics system called Analytic Phonics which is considered slower and tends to place more emphasis on phonemes at the beginning of each word.
In the 1960's an alternative system was trialled in British schools called Initial Teaching Alphabet (ITA) and it was used to teach children up to age-7. The idea was that as children became fluent that they could read clearly and could learn the correct spellings later when they had more confidence. While this worked for some children, for the most part they didn't and there are many adults now who find spelling incredibly difficult.
Your child will pick up words that do not fit the phonics patterns as they go, and the teaching materials at the school are correct. If you would like to work with your child and understand more about what they're doing then why not ask the teacher to explain what phonics system they use and how you can help?