Your question makes a questionable assumption:
Frodo's three hobbit companions in The Lord of the Rings were not yet "of age" (33) when they left to go to Rivendell
This is not entirely true. Pippin was not yet 33, but Merry and Sam both were; the birth years of each of the four Hobbits is given in Appendix C as (years in Shire Reckoning):
- Sam: 1380
- Merry: 1382
- Pippin: 1390
The events of The Lord of the Rings occur in the year S.R. 1419, meaning Sam was 39, Merry was 37, and Pippin was 29.
So I'm going to modify your question slightly:
Why was Pippin allowed to join the fellowship, despite not being of age?
Elrond actually does raise doubts about Pippin, specifically because of his age; he relents largely because Pippin makes it clear that he intends to go anyway, and Elrond clearly doesn't want to imprison him:
In any case, I judge that the younger of these two, Peregrin Took, should remain. My heart is against his going.'
'Then, Master Elrond, you will have to lock me in prison, or send me home tied in a sack,' said Pippin. 'For otherwise I shall follow the Company.'
'Let it be so then. You shall go,' said Elrond, and he sighed. 'Now the tale of Nine is filled. In seven days the Company must depart.'
Fellowship of the Ring Book II Chapter 3: "The Ring Goes South"