Must've read a different book to you.
If I had designed TCAS I would've got it to provide an RA way before a near miss, but then again I didn't.
I have had RAs out of EGLC and seen the conflict aircraft. There was no way it was a near miss. TCAS works on closure rates as follows.
Traffic Advisory (TA) is generated when a predicted aircraft path penetrates the Caution area, 20-48 seconds before entering the Collision Area.
Resolution Advisory (RA) is generated when a predicated aircraft path penetrates the Warning area, 15-35 seconds before entering the Collision Area.
15-35 seconds is plentry of time to take action and has a margin built in to avoid a "near miss".