Not even patrolling. Aimed full speed at Mt Doom in a straight line (according to the proposed map and speed vs stealth hypothesis). It would be reasonable they could be spotted early on, and suddenly the question Sauron has to answer is: "What could a small, fast air force sent directly towards Mt Doom be up to?", to which the answer is obvious. This would make him likely to set up something (interception, barrage...) sufficiently likely to jeopardise the mission. The fact that it is not discussed as a strategy is a much better argument as to why it's a plot hole than any technical point. Still, at that time it is not known what kind of forces stand in Mordor. Besides, the protagonists may not be seeing the task as being so urgent at it looks towards the very end. This makes a prudent approach more reasonable than an all-in decision.

My very personal hunch is that the hobbits (incl. Gollum) are supposed to do this completely by themselves. Many times Gandalf is asked if things will succeed, and many times he worries about their situation and merely hopes they fare well. Still, he refuses to leverage any form of magical or network power to gather any intel — however remote so as not to disclose their location — regarding their progress and simply waits for something to happen. As a plot device, he basically delegates this to Gollum which stands as the guide towards Mt Doom and the Evil that Frodon has to stand against, hence making him stay on the Good side. It is likely that without Gollum, however short the timing (e.g in a supposed fast-flight), Frodon would succumb to the ring.