Still think you need a pull up on the base of Q1 since there is no pull up on U1a. Or you could use a push pull driver. Then it would turn on and off fast.

 

Two problems with that circuit, it is not pole to pole (emitter follower), and you DO need a pull up for that configuration. A pull up resistor is truly not needed for mine, not sure why you are so focused on it. I've been doing this long enough I am positive about what I'm talking about. Generally it is used to make a transistor more immune to noise, a problem I have run into in past.

I'll draw up the conceptual circuit when I have more time.

 

OK, this the equivalent circuit...