That line of reasoning was first argued forcefully in W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205, and the ideas are further discussed in Penelope Maddy, “Believing the axioms”, in two parts, JSL vols. (1988) 52 and 53. Part 1, Part 2. These articles have now a rather large literature of discussion and criticism in the philosophy of set theory. To get started, you might find further resources on the reading list of my recent course NYU Philosophy of Set Theory.

The line of reasoning you mention at the end of your post, firmly in support of large cardinals, was first argued forcefully in

• W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205

and the ideas are further discussed, explained and basically supported in

• Penelope Maddy, Believing the axioms. I, J. Symbolic Logic 53 (1988), no. 2, 481--511.

• Penelope Maddy, Believing the axioms. II, J. Symbolic Logic 53 (1988), no. 3, 736--764.

These articles have now a rather large literature of discussion and criticism in the philosophy of set theory. To get started, you might find further resources on the reading list of my recent course NYU Philosophy of Set Theory. One can now find numerous articles arguing on any given side of each issue.