the greatest poet that ever lived

No, I think you are wrong in that assumption. The phrase is taken to mean "the greatest poet ever, alive or dead."

You are not correct. "Ever lived" is not restricted to dead people. Living is not an "act", it is a state. I have lived in the past; I live in the present; I shall (I hope) live in the future. If we wished to confine the statement to dead poets, we would say "the greatest non-living poet".

I think the issue arises because of the tendency of US English generally to use the present perfect tense (have/has + past participle) far less than speakers of British English. In spoken American English it is very common to use the simple past tense as an alternative in situations where the present perfect would usually have been used in British English.

(US) Jenny feels ill. She ate too much.
(British) Jenny feels ill. She has eaten too much.

(US) TS Eliot is the greatest poet that ever lived.
(British) TS Eliot is the greatest poet that has ever lived.

(US) Seventy five percent of people who ever lived are alive today.
(British) Seventy five percent of people who have ever lived are alive today.

Thanks all~

As to contrex's mention of the US/UK difference, I notice such a thing while watching American cartoons also.

According to contrex, does it mean THAT EVER LIVED is very unlikely to appear in UK English and THAT EVER LIVED can be comprehensible/natural/idiomatic only in US English?

"That ever lived" is somewhat less likely (not "very unlikely") to appear in UK English, and is not comprehensible "only" in US English. Both forms are comprehensible to US and British English native speakers.

Although I have lived, I'm not necessarily dead.

In fact I'm not dead at all.

Thanks all~

I've learned a lot!

An answer in a nutshell might have been formed by recalling the words of a keen observer of precise English usage: "It depends on what the meaning of the word 'is' is."

-I am at all times willing to do anything that seems to me likely to help Monkey, but I have a certain hesitation about the draft letter that you have sent me.

If at the beginning of a reply letter, would Americans use simply SENT for HAVE SENT?