That sounds odd nowadays, but there have been times and places where this was OK. FWIW, David Ben Gurion fought against את in this kind of sentence - not sure why, among all the important wars he fought, he found time for this strange one - and he would have actually preferred your suggestion.
It's even more devilish here, because שותה ends with /e/ which joins the /e/ of את. So even when talking "properly", at a reasonable speed it's hard to distinguish between /shote et hakafe/ and /shote takafe/. Though listening carefully you should hear that in the former the /e/ is longer (two /e/'s).
You’ve probably listened to native speakers of English teaching an English course where they enunciate each word carefully so that listeners will better understand. “I don’t know”, they might say, enunciating the “t”. But in ordinary conversation, native speakers don’t enunciate the “t”, so people taking such a course would have difficulty understanding an ordinarily spoken “I don’t know”.
Similarly in the long run it’s best to hear Hebrew spoken at normal speed and with normal smooshing together of sounds.
Memrise Duolingo is a site where the words are spoken more clearly and there are pauses in between the words. It’s effective and enjoyable.
You're right. אני שותה קפה is how you would say "I drink coffee" or "I'm drinking coffee". However, it is also said quite a lot like in this sentence, depends on context. Some examples where I would use the definite article:
It's a specific cup of coffee I'm talking about. ״אני שותה את הקפה ובא״ - meaning something like "I'm just gonna drink the (cup of) coffee and I'm coming"
It's a specific kind of coffee. I drink my coffee without sugar = אני שותה את הקפה שלי בלי סוכר. I drink the Ethiopian coffee = אני שותה את הקפה האתיופי.
But it is הקפה. Do you perhaps mean why isn't it just הקפה without את? The answer is simple - when the object is definite, both את and ה before the noun must be included -> אני שותה את הקפה - I drink the coffee. That is how Hebrew works.
Or another example: אני אוכל את הלחם - I am eating the bread.
If it were "I drink coffee" then it would be אני שותה קפה. Or if I were to say "I eat bread" אני אוכל לחם.