There are plenty of accents and dialects that 'drop' certain letters or words.
I had my speech 'corrected' so much that I now only speak a 'received' version of my natural speech. This is also slightly to do with the fact that in my adult life I moved around, which knocks the corners off a local accent.
Everywhere I went 'posh' people would tell me 'oh it's not a proper accent they speak here, it's just lazy'. No-one seemed able to define a 'proper accent'.
Now I am back where I belong, and 'local historians' want us to record 'old folk' talking, so our 'lovely local speech' is not lost, dropped letters & ungrammatical constructions and all!
I feel patronised, as if only middle-class people get to decide whose speech is worthy.
Sorry, Franbern, it's a real 'red flag' for me!! And I have lived in areas where the dropped 't' is a feature of the speech.
Of course I don't dispute your right to be irritated by certain ways of speaking.