Não, Sauron não poderia simplesmente estender a mão e pegar o anel.

Acho que a citação para responder à sua pergunta é a que você forneceu (ênfase minha).

He was kneeling in clear sunlight before the high seat. A black shadow seemed to pass like an arm above him; it missed Amon Hen and groped out west, and faded. Then all the sky was clean and blue and birds sang in every tree.
_{The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)}

The world seemed to have shrunk and fallen silent. He was sitting upon the Seat of Seeing, on Amon Hen, the Hill of the Eye of the Men of Númenor...
_{The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)}

O texto que precede sua citação sugere ainda que Sauron estava procurando pelo Anel e não foi capaz de pegá-lo fisicamente

Then at last his gaze was held: wall upon wall, battlement upon battlement, black, immeasurably strong, mountain of iron, gate of steel, tower of Adamant, he saw it: Barad-Dûr, Fortress of Sauron. All hope left him.

And suddenly he felt the Eye. There was an eye in the Dark Tower that did not sleep. He knew that it had become aware of his gaze. A fierce eager will was there. It leaped towards him; almost like a finger he felt it, searching for him. Very soon it would nail him down, know just exactly where he was. Amon Lhaw it touched. It glanced upon Tol Brandir – he threw himself from the seat, crouching, covering his head with his grey hood.
_{The Fellowship of the Ring, Chapter 10 (The Breaking of the Fellowship)}

Esta passagem sugere que, embora Sauron estivesse procurando por ele,

"He knew it had become aware... It leaped towards him... he felt it"

Este trecho discute a busca de Sauron pelo anel, com Sauron se tornando consciente do olhar de Frodo (sugerindo que ele vê alguém procurando) e que a busca de Sauron "pulou em sua direção" e Frodo podia sentir seu olhar.

Isso me leva a concluir que ele teria sido incapaz de agarrar o anel diretamente de Frodo de longe, e só foi capaz de identificar a localização de Frodo. Para, por sua vez, enviar o Nazgûl em busca do anel. Além disso, o "Shadow" era, IMO, uma representação literária para o olhar de Sauron e possivelmente um retrato de seu mal (como a resposta de Blackwood sugere)

A sombra apenas representa Sauron

Eu não encontrei uma discussão deste ponto por Tolkien, mas lembre-se que ele freqüentemente usa a "sombra" como uma metáfora para algo que é sinistro ou maligno. Alguns exemplos estão no Poema do Anel "Na Terra de Mordor, onde as sombras se encontram". e no título do segundo capítulo de O Senhor dos Anéis "A Sombra do Passado".

Às vezes a sombra não é apenas metafórica, mas pode ser vista como uma nuvem ou neblina. A passagem citada na pergunta menciona uma "sombra negra" que é "semelhante e braço". Nós não sabemos se alguém além de Frodo vê este braço.

Quando o Anel é destruído, lemos:

‘The realm of Sauron is ended!’ said Gandalf. ‘The Ringbearer has fulfilled his Quest.’ And as the Captains gazed south to the Land of Mordor, it seemed to them that, black against the pall of cloud, there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.

_{The Lord of the Rings Book Six, Chapter 4: The Field of Cormallen}

Quando Saruman é morto, lemos:

To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing.

_{The Lord of the Rings Book Six, Chapter 8: The Scouring of the Shire}

Ele conseguiu carregar o Anel dos destroços de Numenor.

Though reduced to ‘a spirit of hatred borne on a dark wind’, I do not think one need boggle at this spirit carrying off the One Ring, upon which his power of dominating minds now largely depended.

(Cartas, 211)

Ele teria que viajar para lá como um espírito , é claro.

Denethor laughed bitterly. 'Nay, not yet, Master Peregrin! He will not come save to triumph over me when all is won. He uses others as his weapons. So do all great lords, if they are wise, Master Halfling. Or why should I sit here in my tower and think, and watch, and wait, spending even my sons? For I can still wield a brand.'

(SdA)