I have a client who is sometimes texting me  like "I miss you enormously" bla bla. We just met twice.
It is sweet and can cheer up the day but... just book or shut up instead of sugarcoating
I am down to earth and this kind of text annoys me as it leads to nowhere.
I once texted him back  "write it in my review then" so at least this could somewhat benefit me. But no, he didn't do it.

What would you do with him?
Do you think I could use some flirting to make him book?
Feels like there is a room for something since he is infatuated with me.

Btw, Do I sound like a jaded ungrateful bitch?

I would make it clear in one of several ways;

Ignore unless the message contains information directly relevant to a booking.

Thank him, advise you cannot be pen pals.

Must dash, feel free to get in touch when ready to book (then ignore until he sends a message containing booking information)

I have had repeat bookers who start with a general 'Are you busy/welll/how are you?" message, thinking they have to chat me up. Once I establish they actually want a booking I ask them to 'do me a favour/help me out' then explain in future I need this is your first message or sentence of phone call. Sometimes I provide an example or template.

Some understand, some just carry on. After a few requests like this I just stop responding.

Yes What VC said is the way to deal with him.