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But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
This is the LOTR equivalent of
"ACTUALLY the transporter can do almost anything in Star Trek and should be used constantly to solve every problem and renders every other technology on the show obsolete, including the spaceships themselves" and it is only cute in very small doses.
The original was "why didn't the eagles fly to mordor and drop the ring in the top of the volcano" which was funnier the first time than the thousandth time. You're right that these witty workarounds are only funny the first time and tire quickly.
The answer is that the eagles are theologically angels in LotR and both sapient and very susceptible to the Ring, and having a flying Sauron-replacement is not an improvement.
I thought it was that the eagles were basically demigods and they didn't originally help frodo for the same reason we don't regularly help carry ants back to their mound with food but then after Gandalf came back he was able to convince them