This one has always never made sense to me. Why would Lin Biao ruin everything when Mao was already in failing health and he was the chosen successor? It's always something I found bizarre.
I've heard it suggested, that it was just another paranoid plot by Mao, but to me that doesn't make any sense as well as Lin and Mao were very close allies, saw eye to eye, Mao was quite distressed and depressed afterwards and it severely damaged the credibility of Mao and the Cultural Revolution, which Lin Biao embodied in the eyes of the public.
The whole Lin Biao coup is just so bizarre.
First of all, there has always being some controversy (see Uhalley, Jr. and Jin Qiu 1993) with the official PRC narrative that Lin was launching a coup. Plan 571 was probably never Lin Biao's plan per see, but his son Lin Liguo's. There is a big lack of official documents and materials on Lin's downfall because of Chinese government secrecy on the issue. It could being that Lin never did launch a coup and the government used his son's plan afterwards as evidence that he did to destroy his reputation and powerbase in the aftermath of his death.
But an abortive coup isn't entirely unreasonable either, for one by 1971 there has being rifts between Mao and Lin, and between Lin and other members of the Communist party. During the cultural revolution Lin was at the forefront of the radical movement (he had edited the infamous Red Book of Mao), and viciously persecuted many of his political opponents like the popular Peng De Hui, which alienated him from the moderates.
At the same time, a rift was developing between him and Mao, Mao had disliked comments that Lin made about his wife Jiang Qing. Lin and Jiang had previously had a falling out due to Jiang's attempt to interfere with the army. Mao also feared that Lin's political base in the army was becoming too independent of himself. Mao wanted Lin to make a self-criticism (which was a very common tactic used by Mao to undermine political opponents and a prelude to many's downfall) which Lin refused. Mao also started to shift towards Zhou Enlai's moderates in what could be an attempt to counter Lin influences. Key example of this is the opening of relations with the US.
So basically you had a situation where it's entirely plausible that Lin would be removed and purged. It should be noted that the previous heir apparent to Mao: Liu Shoaqi was purged and died in a pool of his own vomit just a few years prior to this. In fact in the entire history of the PRC it was not until the 1990s that an anointed heir apparent (jiang zemin) to the leader was able to successfully succeed his predecessor and maintain power. Something like 5-6 appointed heirs failed to do so between the 40s and the 90s. So being heir apparent didn't mean that much.
There has being a bunch of alternative theories on what really happened in 1971, they tend to fall into either the "Lin was trying to flee as oppose to launch a coup" camp. But given the circumstances of the early 70s Lin's political position was weakening and either a coup or fleeing is within the realm of plausibility. I suspect we will not know what truly took place until the current Chinese government decide to release documents which would shed light on the incident.