I'm going to be quoting liberally from Akallabêth, one of the last chapters of The Silmarillion. That chapter essentially answers all of your questions.
First off, you're mistaken about why Aman was removed from the world. Aman was never in danger of being invaded by Sauron, but by Ar-Pharazôn the Golden, King of Númenor:
Ar-Pharazôn, being besotted, and walking under the shadow of death, for his span was drawing towards its end, hearkened to Sauron; and he began to ponder in his heart how he might make war upon the Valar.
Thus the fleets of the Númenóreans moved against the menace of the West[.]
In fact Sauron wasn't even part of the fleet that moved against the Valar; we know this because he's sitting in the Númenórean temple laughing at the foolish mortals:
And Sauron, sitting in his black seat in the midst of the Temple, had laughed when he heard the trumpets of Ar-Pharazôn sounding for battle[.]
It was also never Sauron's intention to wage war against the Valar. Sauron isn't stupid; he knows that he can't destroy them, and no force exists on Middle-Earth that could do them any serious harm1. In fact Sauron seems pretty happy to leave Aman to the Valar; he just wants dominion over Middle-Earth. Sauron's goal with Númenor is much pettier:
For Sauron himself was filled with great fear at the wrath of the Valar, and the doom that Eru laid upon sea and land. It was greater far than aught he had looked for, hoping only for the death of the Númenóreans and the defeat of their proud king. And Sauron, sitting in his black seat in the midst of the Temple, had laughed when he heard the trumpets of Ar-Pharazôn sounding for battle; and again he had laughed when he heard the thunder of the storm; and a third time, even as he laughed at his own thought, thinking what he would do now in the world, being rid of the Edain for ever[.]
Sauron basically goaded Ar-Pharazôn into fighting an unwinnable battle, just so the Númenóreans would be destroyed. Why? Partly because of their allegiance with his hated enemies, the Elves and the Valar, and partly because of the butt-whooping they had given him after he revealed the One Ring:
And Sauron hated the Númenóreans, because of the deeds of their fathers and their ancient alliance with the Elves and allegiance to the Valar; nor did he forget the aid that Tar-Minastir had rendered to Gil-galad of old, in that time when the One Ring was forged and there was war between Sauron and the Elves in Eriador.
But that still raises the question: why did Eru remove Aman in the first place? Why was it such a big deal that mortal men walked in the Undying Lands?
Akallabêth does us a favour by answering this question, but also showing why the Númenóreans didn't pay attention to it. Long before Ar-Pharazôn's rule, the Númenóreans spoke to some Elves concerning death and the Undying Lands:
And the Númenóreans answered: 'Why should we not envy the Valar, or even the least of the Deathless? For of us is required a blind trust, and a hope without assurance, knowing not what lies before us in a little while. And yet we also love the Earth and would not lose it.'
Then the Messengers said: 'Indeed the mind of Ilúvatar concerning you is not known to the Valar, and he has not revealed all things that are to come. But this we hold to be true, that your home is not here, neither in the Land of Aman nor anywhere within the Circles of the World. And the Doom of Men, that they should depart, was at first a gift of Ilúvatar. [...] The will of Eru may not be gainsaid; and the Valar bid you earnestly not to withhold the trust to which you are called, lest soon it become again a bond by which you are constrained. Hope rather that in the end even the least of your desires shall have fruit. The love of Arda was set in your hearts by Ilúvatar, and he does not plant to no purpose. Nonetheless, many ages of Men unborn may pass ere that purpose is made known; and to you it will be revealed and not to the Valar.'
This isn't a terribly satisfying answer, from the point of view of the Númenóreans; it essentially boils down to "don't sail West because Eru doesn't want you to be immortal." This is even more of a non sequitur when you remember that just being in Aman isn't enough to make you immortal2 (which of course the Númenóreans don't realize). A better answer comes earlier in the chapter, when the Ban of the Valar is first discussed.
But the design of Manwë was that the Númenóreans should not be tempted to seek for the Blessed Realm, nor desire to overpass the limits set to their bliss, becoming enamoured of the immortality of the Valar and the Eldar and the lands where all things endure.
The Ban of the Valar is again discussed in Letter 131 (emphasis Tolkien's):
[The Númenóreans'] long life is their undoing - or the means of their temptation. Their long life aids their achievements in an and wisdom3, but breeds a possessive attitude to these things, and desire awakes for more time for their enjoyment. Foreseeing this in pan, the gods [Valar] lay a Ban on the Númenóreans from the beginning: they must never sail to Eressëa, nor westward out of sight of their own land. In all other directions they could go as they would. They must not set foot on 'immortal' lands, and so become enamoured of an immortality (within their world), which was against their law, the special doom or gift of Ilúvatar (God), and which their nature could not in fact endure.
This seems to suggest that the Ban was in place to prevent Men from wasting their lives pining for something they'll never achieve. Combined with the cryptic answer of the Elves, above, the reason seems to be that Eru has some design for Men beyond the Second Music, and He needs there to be good men to fulfil his purpose.
1 Shamshiel points out in comments that the Númenóreans could have done significant damage in Aman, had they been allowed to land. That's likely true, but mostly they would have killed Elves and violated the sanctity of the land. They wouldn't be able to significantly harm the Valar personally, except insofar as the death and destruction would hurt them emotionally. Tolkien himself admits the possibility of damage in Letter 131, though he doesn't address exactly how ruinous it would have been:
[T]he Númenóreans directed by Sauron could have wrought ruin in Valinor itself.
2 From Akallabêth, again:
And were you so to voyage that escaping all deceits and snares you came indeed to Aman, the Blessed Realm, little would it profit you. For it is not the land of Manwë that makes its people deathless, but the Deathless that dwell therein have hallowed the land
3 I've preserved the phrasing that exists in my copy of Letters.