Ashikaga Yoshiaki was a mean, cruel, and feeble minded shogun. He was not not up to the job. Sure, he had better claim to succeed than his cousin Yoshihide. In fact, the Rokkaku and the Asakura did not help him. The only person who would help Yoshiaki to Kyoto and take the title of Shogun was Nobunaga. Nobunaga's objective was not to resurrect the old regime, but to legitimate his own goal, the realm. Nobunaga eventually banished Yoshiaki in 1573. In the end, Yoshiaki was a king without a kingdom.
This passage from the Chronicle of Lord Nobunaga explains other families did not bother with Yoshiaki. They probably knew what Nobunaga and others knew, Yoshiaki was not guy.
(p.115) "The kubo, Lord Ichijion (Ashikaga Yoshiaki), had appealed to Sasaki Jotei for support, Jotei, however, failed to heed his call. Next, proceeding to Echizen, Lord Ichijion asked Asakura Sakyo no Daibu Yoshikage for assistance, but that too, did not bring him any closer to Kyoto. Finally, he decided to put his trust in Kazusa no Suke Nobunaga. His message to that effect was transmitted to Nobunaga through Hosokawa Hyobu no Daibu (Fujitaka) and Wata Iga no Kami (Koremasa)."
Tenka no tame!