TL;DR penny-anna is wrong, man/men refers to males, not the race

Tolkien capitalise the m when speaking of the race Men. "Man" refers to the race of full sized humans and "man" refers to a male.

Chapter Treebeard:

‘We always seem to have got left out of the old lists, and the old stories,’ said Merry. ‘Yet we've been about for quite a long time. We’re hobbits.’ ‘Why not make a new line?’ said Pippin. ‘Half-grown hobbits, the hole-dwellers. Put us in amongst the four, next to Man and you've got it.’

Man is capitalised as they speak about the race of men

Chapter Shelob's lair:

But that desire was yet far away, and long now had she been hungry, lurking in her den, while the power of Sauron grew, and light and living things forsook his borders; and the city in the valley was dead, and no Elf or Man came near, only the unhappy Orcs.

Again, they speak of the race

Chapter The Battle of Pelennor Fields (Witch-King speaks of his prophecy):

Hinder me? Thou fool. No living man may hinder me!

Man is not capitalised as the prophecy was about males, not the race of men, which is a category Éowyn would have fallen into.

As summarised in the 50th anniversary edition :

... The result, nonetheless, still includes many variations in capitalization, punctuation, and other points of style. Not all of these are erroneous: they include words such as Sun, Moon, Hobbit, and Man (or sun, moon, hobbit, man), which may change form according to meaning or application, in relation to adjacent adjectives, or whether Tolkien intended personification, poetry, or emphasis. His intent cannot be divined with confidence in every case. But it is possible to discern Tolkien’s preferences in many instances, from statements he wrote in his check copies of The Lord of the Rings or from a close analysis of its text in manuscript, typescript, proof, and print. Whenever there has been any doubt whatsoever as to the author’s intentions, the text has been allowed to stand.