They don't normally need to. As Boromir comments at The Council of Elrond:
Some said that it could be seen, like a great black horseman, a dark shadow under the moon. Wherever he came a madness filled our foes, but fear fell on our boldest, so that horse and man gave way and fled.
For the start of their journey to the Shire they actually remained invisible, as we learn in The Hunt for the Ring (published in Unfinished Tales):
The Lord of Morgul therefore led his companions over Anduin, unclad and unmounted, and invisible to eyes, and yet a terror to all living things that they passed near. It was, maybe, on the first day of July that they went forth. They passed slowly and in stealth, through Anórien, and over the Entwade, and so into the Wold, and rumour of darkness and a dread of men knew not what went before them. They reached the west-shores of Anduin a little north of Sarn Gebir, as they had trysted; and there received horses and raiment that were secretly ferried over the River. This was (it is thought) about the seventeenth of July. Then they passed northward seeking for the Shire, the land of the Halflings.
Indeed the primary purpose of wearing cloaks appears to be to enable them to interact with people, in addition to the well-known questioning of Hobbits in the Shire, The Hunt for the Ring notes a number of other such interactions:
- With Saruman
- With Wormtongue
- With some spies of Saruman's in Minhiriath.
As for the effectiveness of invisibility in taking the Ring from Frodo on Weathertop, you're seeing a plot-hole where none exists. The Nazgul's primary weapon isn't invisibility, it's fear, again from The Hunt for the Ring:
...so great was the terror that went with them (even invisible and unclad)...
This extract even goes so far as to suggest that the fear they cause is even reduced when they are invisible.
We see this in the Weathertop scene, where the book describes the Hobbits' reactions:
Terror overcame Pippin and Merry, and they threw themselves flat on the ground. Sam shrank to Frodo's side. Frodo was hardly less terrified than his companions; he was quaking as if he was bitter cold...
So the Nazgul hardly need to be invisible, they just need to scare the living bejeezus out of people, and there is a suggestion that they can more effectively do so if they can be seen.