Ndtyhij

He was certainly more resistant to the Ring than any other character in The Lord of the Rings. He could have handled the Ring with greater ease throughout the journey, especially in Mordor, as compared to Frodo.

Does this idea not come to anyone's mind or is there some deeper reason?

The Council of Elrond discusses asking Bombadil to help:

[Elrond]: But I had forgotten Bombadil, if indeed this is still the same that walked the woods and hills long ago, and even then was older than the old. That was not then his name. Iarwain Ben-adar we called him, oldest and fatherless. But many another name he has since been given by other folk: Forn by the Dwarves, Orald by Northern Men, and other names beside. He is a strange creature, but maybe I should have summoned him to our Council.'

'He would not have come,' said Gandalf.

'Could we not still send messages to him and obtain his help?' asked Erestor. 'It seems that he has a power even over the Ring.'

'No, I should not put it so,' said Gandalf. 'Say rather that the Ring has no power over him. He is his own master. But he cannot alter the Ring itself, nor break its power over others. And now he is withdrawn into a little land, within bounds that he has set, though none can see them, waiting perhaps for a change of days, and he will not step beyond them.'

'But within those bounds nothing seems to dismay him,' said Erestor. 'Would he not take the Ring and keep it there, for ever harmless?'

'No,' said Gandalf, 'not willingly. He might do so, if all the free folk of the world begged him, but he would not understand the need. And if he were given the Ring, he would soon forget it, or most likely throw it away. Such things have no hold on his mind. He would be a most unsafe guardian; and that alone is answer enough.'

He would neither venture from his land nor be reliable enough to carry the Ring to Mordor. Gandalf says that, if asked, he would not have even come to Rivendell. But if begged, he might have taken the Ring to hold, but "if he were given the Ring, he would soon forget it, or most likely throw it away." He just doesn't care about the Ring and Sauron the way everyone else does, and he can't be made to.

I might put it this way. The story is cast in terms of a good side, and a bad side, beauty against ruthless ugliness, tyranny against kingship, moderated freedom with consent against compulsion that has long lost any object save mere power, and so on; but both sides in some degree, conservative or destructive, want a measure of control. But if you have, as it were taken 'a vow of poverty', renounced control, and take your delight in things for themselves without reference to yourself, watching, observing, and to some extent knowing, then the question of the rights and wrongs of power and control might become utterly meaningless to you, and the means of power quite valueless. It is a natural pacifist view, which always arises in the mind when there is a war.

^{_{The Letters of J.R.R. Tolkien 144: To Naomi Mitchison. April 1954}}

If he wouldn't travel outside his land to come to Rivendell, and if he would forget the Ring or throw it away if he had it -- the Ring that was trying to return to Sauron -- could he have possibly been persuaded to go to Mordor or trusted to complete the mission?

Bombadil could not have done it.