Personally, I think Tao Qian's third invitation to Xuzhou is sincere. He really wants Liu Bei to take over Xuzhou, because he also knows that his time is running out, and there is no more suitable person in Xuzhou to take over. What's more, he admired Liu Bei's personality, because he invited so many governors to save Xuzhou, and no one came, but Liu Bei, who he didn't invite, brought thousands of people to help him. I think this is where Tao Qian values Liu Bei.

Tao Qian, originally a secretariat of Xuzhou, is known for being close to the people. Everyone's evaluation of him is benevolence for the people, but he made a mistake, because he wanted to make friends with Cao Cao, which led to Xuzhou being above the gun. He didn't expect that in order to make friends with Cao Cao, he indirectly killed Cao Cao's father, which made Cao Cao find a legitimate reason to attack Xuzhou, and Tao Qian also knew that Cao Cao was bound to seize Xuzhou, so he defended Xuzhou to the death and refused to surrender.

However, Tao Qian's personal strength was not enough to deal with Cao Cao, so he wrote to invite several governors around him to help, while Yuan Shao brothers and others were unwilling to be enemies of Cao Cao to help a weak Tao Qian. Only Liu Bei, who didn't receive the invitation but just got the news, came to help and let Cao Cao withdraw temporarily, so for Tao Qian, regardless of Liu Bei's purpose, he really did it for Tao Qian.

And Tao Qian, in this case, immediately made a decision to let Liu Bei take over Xuzhou to resist Cao Cao's attack, but Liu Bei did not agree at first, until the third time, Tao Qian pointed out Liu Bei's idea, and Liu Beicai agreed to Tao Qian. So from a personal point of view, I think Tao Qian must be sincere, not to mention the nature of Tao Qian, and more importantly, for the safety of Xuzhou, it was also the best choice at that time to give Xuzhou to Liu Bei.