Basically: Japan has tried to get rid of Kanji for convenience several times since the mid-19th Century. However, the first attempts which were mainly supported by the idea that regular civilians had low kanji literacy, amounted to nothing as kanji literacy among regular Japanese civilians was higher than expected.
For a while in the 20th century the 常用漢字 was actually named 当用漢字 or "provisional kanji list" as there was yet another push for education reforms that would gradually diminish kanji use. This was especially poignant before the arrival of personal computers, as there were no convenient input methods for kanji with typewriters.
But this was halted, once again, as personal computers provided a convenient and easy way for typing Kanji. Which meant that there was no real need to stop using it.
Ultimately, Japan hasn't abandoned Kanji because it hasn't been necessary. Most people already know how to read it, and it provides easier access and understanding of their historical texts.
Personally, I also think Kanji provides several advantages, such as an immediate understanding of vocabulary based solely on its kanji, or the ability to transmit more information in fewer characters.