I know this seems awful to think about as decent people, but consider what Mordor would have done in the situation of the good guys. Who is to say that after Frodo was stabbed, that they wouldn't just leave him to die and give the Ring to someone stronger like Aragorn? Staying with Frodo would endanger the lot of them, especially if he was won over by the wound and fell to the will of the Enemy. One would have thought that Sauron, who was so power hungry that the idea of someone wanting to destroy the Ring rather than use it NEVER occurred to him till Frodo claimed it on Mount Doom and the whole plan of the Council of Elrond was exposed to him, would have assumed that Strider and the three other Hobbits would leave Frodo to die and run with the Ring themselves in hopes of defeating Sauron and using the Ring themselves. That they would run the risk of Frodo falling and all being lost wouldn't seem like something that Mordor would think of.