Why did Homer include types of help from the immortals and Odysseus himself instead of presenting only one way to overcome obstacles?

By including both help from the gods and instances of Odysseus helping himself with his wit and strength, Homer is able to show that Odysseus is a hero favored by the gods and able to allow Odysseus to escape situations that would not have been escapable without divine intervention.
If Homer had allowed Odysseus to overcome all obstacles with help from the gods, Odysseus would have been lucky, but he would not have been a hero. If Homer allowed Odysseus to succeed without help from the gods, Homer would have been guilty of minimizing the role that the gods played in the lives of Homer's audience. Many of the situations that Odysseus found himself in would have been impossible to escape without help from the gods, especially because his immediate situation resulted from angering Poseidon, a major Greek god.
Odysseus offends Poseidon when he blinds Polyphemus, and, as a result, Odysseus is sent on a voyage that keeps him away from home for decades. Because a god—Poseidon—is the cause of Odysseus's long misfortune, he needs divine intervention to overcome the problem. Odysseus cannot stand against a god and succeed. He needs other gods.
Luckily, Athena is willing to intervene for him. Her patronage causes Zeus and Hermes to assist him as well, getting him out of many spots that might have otherwise resulted in his death. For example, Hermes tells Calypso that Zeus wants Odysseus to journey home. If he had not done this, Odysseus might have been stuck with Calypso forever. The assistance of the gods helps Odysseus overcome obstacles that a mortal man would not have been able to.
But Odysseus does not always need the gods to escape trouble. For example, when he and his men escape Polyphemus, it is Odysseus who plans to tie the men to the bellies of sheep, to get the Cyclops drunk, and to give him a false name so that the other Cyclops will not give chase. His cleverness and strength as he navigates the difficult challenges are what make Odysseus himself a hero and a worthy king of Ithaca. In fact, his own resilience and skill also make him someone who is worthy of receiving help from the gods. Homer uses Odysseus's ability to get out of danger and make his way home to Ithaca to show the reader that Odysseus is a hero who deserves help from the gods.