It is known to all educated people -- and probably quite a few others --
but few act like it when it comes to dying.
That we have a life after death is general knowledge handed down from ancient times in our world, yet how many people believe it enough to act like they do? People today fail to believe it for many reasons:
Probably biggest is that it is known only from ancient records, but no one has any personal experience of life after death or knows anyone who did. Is this handed-down lore something you would bet your life -- literally -- on? For most people today, no. Likewise in Middle-earth.
...the Valar were not permitted to take from them the Gift of Men (or the Doom of Men, as it was afterwards called).
"The Doom of Men" -- is there anything in that which suggests that the Edain were any different than us?
Even people who do believe in a life after death, even people who believe that that future life will be much better than what we have here, are rarely in a hurry to depart. We are bound to the here-and-now by our daily affairs ("Little Sarah turns three in just two weeks, I love watching her grow up!") or just a love of the world we find ourselves in. Likewise in Middle-earth, even among the Elves. "There are countless things still to see in Middle-earth, and great works to do...."
...the Elves remain until the end of days, and their love of the Earth and all the world is more single and more poignant therefore, and as the years lengthen ever more sorrowful.
While the Elves may have had an intellectual knowledge of Elvenhome, it was not an emotional one, though that could be awakened. After his first glimpse of the Sea, Legolas says:
'Look!' he cried. 'Gulls! They are flying far inland. A wonder they are to me and a trouble to my heart. Never in all my life had I met them, until we came to Pelargir, and there I heard them crying in the air as we rode to the battle of the ships. Then I stood still, forgetting war in Middle-earth; for their wailing voices spoke to me of the Sea. The Sea! Alas! I have not yet beheld it. But deep in the hearts of all my kindred lies the sea-longing, which it is perilous to stir. Alas! for the gulls. No peace shall I have again under beech or under elm.'
In most elves, the sea-longing has not yet been awakened, and in Men it does not exist at all.
Finally, even if an elf or a man was convinced that a better life after death was possible, there was judgment by Mandos first, and that was not something to be looked forward to, even for someone who had no guilt:
'Ye have spilled the blood of your kindred unrighteously and have stained the land of Aman. For blood ye shall render blood, and beyond Aman ye shall dwell in Death's shadow. For though Eru appointed to you to die not in Eä, and no sickness may assail you, yet slain ye may be, and slain ye shall be by weapon and by torment and by grief; and your houseless spirits shall come then to Mandos. There long shall ye abide and yearn for your bodies, and find little pity though all whom ye have slain should entreat for you.
Who, believing in a better life after death, is eager to rush towards Mandos' judgment?
The Men and Elves we read about in LotR and Silmarillion are people like us, and react to life (and death) pretty much as we do, and do so for pretty much the same reasons. They may have lore or book-learning which says that death is nothing to fear, but they act otherwise.
ADDED: In the comments @Graham raised a good point about the immortal elves being positive proof of something beyond what we, ourselves, know. It bears consideration, but, I think, ultimately fails.
First, consider Men of the Third Age. They know of the Elves...or do they? There's the Last Alliance of Elves and Men which at the time of LotR was 3000 years in the past...about as long ago as The Odyssey or Exodus which report on similarly ancient events. Men in the late Third Age simply never meet or deal with Elves (and those who do are disbelieved.) Ted Sandyman, who probably correctly represents the average Hobbit's perceptions, for instance:
'Queer things you do hear these days, to be sure,' said Sam.
'Ah,' said Ted, 'you do, if you listen. But I can hear fireside-tales and children's stories at home, if I want to.'
...'And I've heard tell that Elves are moving west. They do say they are going to the harbours, out away beyond the White Towers.' Sam waved his arm vaguely: neither he nor any of them knew how far it was to the Sea, past the old towers beyond the western borders of the Shire. But it was an old tradition that away over there stood the Grey Havens, from which at times elven-ships set sail, never to return.
'They are sailing, sailing, sailing over the Sea, they are going into the West and leaving us,' said Sam, half chanting the words, shaking his head sadly and solemnly. But Ted laughed.
'Well, that isn't anything new, if you believe the old tales. And I don't see what it matters to me or you. Let them sail! But I warrant you haven't seen them doing it; nor anyone else in the Shire.'
Eomer of Rohan sees nearby Lothlorien as legend:
'No,' said Aragorn. 'One only of us is an Elf, Legolas from the Woodland Realm in distant Mirkwood. But we have passed through Lothlórien, and the gifts and favour of the Lady go with us.'
The Rider looked at them with renewed wonder, but his eyes hardened. 'Then there is a Lady in the Golden Wood, as old tales tell!' he said. 'Few escape her nets, they say. These are strange days! But if you have her favour, then you also are net-weavers and sorcerers, maybe.'
...'Halflings!' laughed the Rider that stood beside Éomer. 'Halflings! But they are only a little people in old songs and children's tales out of the North. Do we walk in legends or on the green earth in the daylight?'
For Men who were not loremasters, Elves were the stuff of legend, and consequently of no aid in understanding their own Death.
To be sure Numenor, to start with, anyway, knew the Elves:
The realm of Númenor endured to the end of the Second Age and increased ever in power and splendour, and until half the Age had passed the Númenoreans grew also in wisdom and joy. The first sign of the shadow that was to fall upon them appeared in the days of Tar-Minastir, eleventh King. He it was that sent a great force to the aid of Gil-galad. He loved the Eldar but envied them. The Númenoreans had now become great mariners, exploring all the seas eastward, and they began to yearn for the West and the forbidden waters; and the more joyful was their life, the more they began to long for the immortality of the Eldar.
They knew the Elves, but envied their bodily immortality. In spite of their direct knowledge that there is more in heaven and Earth than we imagine, they still came to fear the own deaths. Perhaps a race of Vulcans would conclude with perfect logic that if Elves existed and were immortal, then Men had immortal souls. Perhaps. (But is that really logical?)
In the 'real' event, the Numenorians feared death and envied immortality on Earth in spite of what they saw. Regardless of what they should have concluded, they did not.
It appears that sharing a World with Elves does not do much to remove people's fear of dying.
