The strap and clip have probably been replaced. When something removeable like a strap or hangtag or even a D-ring doesn't match the rest of the bag, that doesn't prove the bag's fake, just that the part may have been replaced at some time.
The missing under-flap pocket is a bigger problem. If the bag's fake, that's the only proof of it, and remember, there's always a remote possibility that Coach made a similar style to the Convertible Clutch but without the pocket under the front flap. I haven't found any in my catalogs but only have a few for the entire time period from the mid-70s to the late 1980s. There don't seem to be any other obvious problems, and you've owned other versions of that style, I haven't. You're probably more qualified to say what's wrong about it than I am.
Thankyou Hyacinth

I appreciate your thoughtful response. I have seen and had some
Bags that have had replacement straps though it seems a shame
to add such an icky plastic replacement to a vintage bag. It certainly was the strap and clips that made me suspect the bag overall.
As far as the bag goes, You're right, I've owned quite a few versions of this bag - I will double check into some of the older NYC Convertible Clutch's I've owned to see if any have missed that inside pocket. If so, I'll feel good about the bag (but not about the cheap strap). However, If there is any chance of the bag being fake it's not a risk I'll wager.
I'll let you know after I check.
Thankyou again Hyacinth!