In the sentence "you have a problem", "have" is the main verb. When reduced to the clitic "'ve", it becomes a weak form and is only expected to be used as an auxiliary verb. These types of verbs must be followed by the main verb. "a" is not a verb. Thus, we insert "got".

If we do not insert "got", the stress in the sentence moves and it sounds overly affected.

I'm not too sure, but I think "be" ("is", "are") is the only verb that can be contracted and still remain a main verb. I'm not too sure why.

English is weird. It can be understood through tough thorough thought, though

You've mail!

I've a solution!

"You've a problem" actually still works, but it's an older way of saying it

Yes, English is weird.

'Bough' and 'cough' are not pronounced the same. 'Bough' and 'bow' are pronounced the same. 'Knee' and 'Leigh' are pronounced the same. 'Neigh' and 'nay' are pronounced the same. 'Polish' (the nationality) and 'polish' (as in what you do to a metal object) are not pronounced the same. 'Tear' (as in to rip) and 'tear' (as in to cry) are spelled the same, but not pronounced the same. Other words which are spelled the same, but pronounced differently:

resume / resume present / present record / record close / close use / use live / live

You've a problem is proper English as well, albeit more often used in the UK than in NA, feel free to use it!

Tom Scott has a great video on contractions.

https://youtu.be/CkZyZFa5qO0

Yeah but that’s not English only. Try saying “de el” in Spanish and it sounds super wrong, for similar reasons. Sometimes contractions kill what they replace

It depends on your emphasis. "You HAVE got a problem there," doesn't sound weird when you emphasize the have. You've a problem doesn't sound weird, just a bit British.

Tom Scott recently did a video about this: https://www.youtube.com/watch?v=CkZyZFa5qO0

No, this is a problem with the language in general.

I actually think that 'you have a problem's and 'you have got/you've got a problem' are subtely different in meaning. If someone has something, they may have had it all along. It sort of has an ongoing ontological quality. If someone has got something, it implies that they got it at some point in time. I think.

People do say that though.

You’ve got mail!

Goodbye!

"you have got a problem", which isn't wrong

Can someone explain to me, why isn't it "you have gotten a problem"?

It also has secret rules look up adjectives ordering and vowel ordering.

Y'all crack me up with many of these comments!

English has quite many problems in a lot of components. It is not phonemic and is very complex. That is one of the main reasons I got interested in learning Esperanto. It is simple, phonemic and planned to be good.

Bonan tagon al ĉiuj esperantistoj ĉi tie.

Contractions are very regional and a product of spoken English which varies quite a lot from place to place. For example, I use contractions that I don’t see people around me using like: y’all (plural 2nd person pronoun that’s missing in official English speech; verbs are conjugated the same as 2nd person singular forms), shouldn’t’ve, gonna (going to/ going to want to), wanna (want to), that’re, then’ll, then’re etc.