Nobunaga was never offered to become Shogun; he was offered by the Emperor to become kanrei - essentially vice Shogun under Ashikaga Yoshiaki, whom Nobunaga himself had restored in his position.
Yoshiaki was the last of the Ashikaga Shoguns, whose title reflected the prestige but not the actual power of the title. After the battle of Okehazama, in which the Imagawa march to Kyoto was stopped on its tracks by Nobunaga, Yoshiaki decided to ask him (Nobunaga) to restore him in power. Nobunaga was quick to see the opportunity and to make the most of it, as it essentially legitimized his moving in to Kyoto and taking over, including the support of the Imperial court and the (Kyoto based) nobility of the country.
After, he "re-instated" Yoshiaki, however, their relationship started turning sour, as it became abundantly clear to Yoshiaki, the Emperor and everyone else that Nobunaga intended to rule alone. His refusal of teh title of kanrei, was a political decision and was not related to anylineage issue; he was just waiting essentially to defeat all the other contesting clans by which point he would have done away with Yoshiaki and the Emperor, as Ieysau later on did when he became Shogun. Yoshiaki and the Emperor did get the "message" of Nobunaga's refusal and were not at all pleased.
http://www.samurai-archives.com/nobunaga.html#3
The person who had ancenctry problems was Nobunaga's able general of humble origins (from a peasant, ashigaru, family) Toyotomi (Hashiba) Hideyoshi. After the murder of Nobunaga by Michuhide, Hideyoshi defeated the murderer (as well as a few other contesting Oda generals) and took over the Oda clan and with it the rulership of Japan. He accepted the title of "Kampaku" ie imperial regent - essentially being Shogun in all but name due to the lineage issue.
http://www.samurai-archives.com/hideyoshi.html
Hideyoshi, left a son, Hideyori who could become Shogun through tracing noble lineage from him mother, Yodo-Gimi iirc. Hideyoshi left a council of the five most powerful Daimyos that would act as regents from his untimely death till his son would come of age. One of them was Ieyasu - the most powerful of the lot, who immediately after Hideyoshis death started political maneuvers to bring as many Daimyos to his side against the Hideysohi lineage. This building of tension led to the monumental battle of Shekigahara in which the Tokugawa gained victory and with it the Shogunate.
Hideyori and his mother still lived on however in Hideyoshi's stronghold, the Osaka castle, that became the hub of anti-Tokugawa factions that rallied around Hideyoshi's heir whom they saw as the ligitimate heir. This led to two more battles - essentially large-scale sieges of the Osaka castle. the second one was conclusive and Yodeyori and his mother commited suicide.
Hideyori by that time also had a young son, whom Ieysau personally order killed, making absolutely certain that there would be no more rallying point of a rivalling dynasty to his. This, as ruthless as it was, made lots of sense as in faudal japan loyalties were largely personal.
Politics and dynastic politics mattered a lot during the period and it would be nice to see that in the game.
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