No, the evidence suggests otherwise...
First of all, I notice that what you are using here is not from Mineta's testimony under oath, but merely an interview for the 'American Academy of Achievement'.
This interview is about his life, and although factual, it is also somewhat colourful, and Mineta will be aware that it has to be 'interesting'.
So it's understandable that Mineta adds a little flavour... do you imagine him really saying to his driver "Is there something wrong with this picture? We are driving in, and everybody else is running away." ?
Now, I notice you don't quote anything further from this interview of Mineta. Why not?
Well, according to your timeline, it's known at this point that the Pentagon has been hit at least 10 minutes earlier, so let's look what Mineta says happens next - from his interview which you snipped :
"So I went into the White House and someone said, "You have to be briefed by Dick Clark in the Situation Room." So I went in there, he talked to me for four or five minutes, and he said, "You have got to go to the PEOC." I said, "What's the PEOC?" He said, "That's the Presidential Emergency Operations Center." I said, "I don't know where that is or what it is." There was a Secret Service agent standing there, says, "I will take you." Well, it's that bunker that's way under the White House."
So why did Dick Clark, the counter terrorism chief, NOT mention to Mineta that the Pentagon has been attacked?
In the situation room, for four or five minutes and no mention of the attack on the Pentagon? which - according to you - was known to have happened 10 mins. before Mineta arrived + 5 mins. talk with Clark = 15 minutes earlier.
So, it's clear that the Pentagon had not yet been hit.
There is no reason to think that Mineta's timeline is incorrect, he was most likely to be in the PEOC well before the Pentagon was hit, and so was Cheney.