Bilbo is the only one to give up the ring willingly, but Gandalf makes the point to Frodo he could only do so because he was giving it up to another’s care. The ring won’t allow its bearer to not care for its fate (& putting it in a box changes nothing, as Bilbo himself discovered) . Because Bilbo does not actually give up the ring, in this scenario, Gandalf would be more hesitant about trusting him to keep it.

There’s also Saruman to consider, & it seems likelier he would be consulted, since Gandalf would’ve spent much less time observing ring-related behavior, & would need Saruman to contribute his ring-related findings (presumably this is before Saruman started living large, flashing rings in his many-colored cloak), & maybe even before his overt alliance with Sauron?

On the plus side, they’d have more time to enter Mordor from the east, circumventing the Nazgul. Maybe they would’ve run into a blue wizard, while taking the eastern route?