No. I think by your second-last sentence you mean '...the symbols agreed by the International Phonetic Association are designed to represent the sounds of all languages'. But those few symbols, with added diacritics, can represent a few thousand 'ideal' sounds. The human buccal tract is capable of producing an unlimited number.
Look up 'allophones'. English speakers represent the phonemes of English by producing various allophones. That process can be called 'English phonetics'.
I'm not sure what the first sentence means, but there is nothing wrong with the phrase 'English phonetics and phonology'.
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