For example, The Tale of Genji was written in the middle of the Heian era when a lot of different literary experimentation was going on, and it took time for it to be recognized.

Really? My impression had been that Genji was recognized almost immediately as one of the great works of Heian literature, based on the profusion of manuscripts prepared in Shikibu's time, the countless imitators, the testimony of the Sarashina Nikki, the commentaries prepared not too long afterwards, and in particular, the very high regard of Fujiwara no Teika, one of the most important literary figures for centuries (I may be biased, since I wrote the Wikipedia entry on Teika), who worked on the manuscript.