The point is:
"есть" is used in possession constructions when the existence is emphasized, and "есть" is omitted when the subject of possession is emphasized.
So, we say "У меня есть велосипед" when we want to emphasize it exists in our possession, something like "I do have a bicycle".
And we say "У меня большой велосипед" when it is obvious I have a bicycle, and I just want to specify what kind of it I have: "The bicycle I have is big".
Note, it's also possible to say "У меня есть большой велосипед", but the meaning is slightly different, now I am accentuating the fact I have this thing: "I do have a big bicycle".

So, when it is not known if the possessor has something or not, use "есть". When it is already known that the object exists, and you only want to provide some details on it, do not use "есть". That's why I did not want to overcomplicate the things.

When talking about parts of body, in 99% of cases, it is obvious that you have it. That is why, we say
У меня длинные волосы.
У меня зелёные глаза.
У меня сильные руки.
У меня прямой нос.
etc.

The example you provided (У меня есть глаза) is not often heard in real speech. It can work in some specific situations when you need to emphasize the fact you HAVE eyes.
I can think of an example of course:
- Ты видел это?
- Конечно видел! У меня же есть глаза!
But it is still a specific context.

Therefore, I believe, the rule I proposed above may work for a beginner in simple phrases.

Moreover, when talking about a disease, we do not normally use "есть", as we usually emphasize WHAT DISEASE we have, not just the fact we HAVE it. Normally, it does not make sense to be proud of possessing a disease
So, we say "У меня ангина" (I have tonsillitis).

But again, in some specific situations we may need to emphasize the fact of existence. Imagine:
- Зачем тебе эти таблетки? У тебя же нет ангины.
- Неправда, у меня есть ангина.