Along the same thought as Dundee's statement above, are you sure it was a full cert?
The few of the original issued birth certs that I have from mine and hubbys relatives possessions ,such as hubby's grandparents are short ones. None of mine and hubby's families had their long certs that I know of.
I would take a good guess that most people back in the first half of the 1900s only ever had their short ones issued to them.
The short ones were very low cost , or free I believe, where you had to pay for the longer ones.
One of our old( hubby's grandfather) original short cert issued at his birth ( in1909) , the writing on the back says to not to exceed 3 pence. The short ones were not entirely free , but would have cost far less than a full one )
My own original issued one at my birth is a short one too, and I was born way, way into the latter half of the 1900s

( I guess my parent's were too tight to pay for the long one too, or really didnt see any need for a long one that would have cost them money) .
I have never needed my long one for anything , driving licences, emigration , passport applications both , Aus and British ones, and green cards in Asia. I used my Short one on everything that needed a birth cert.
The only one exception that I ever needed a long cert was around 15 years ago , when I decided to send off for my 1st Irish passport ( as my Irish citizenship is through my Irish born mother, I needed my long cert to show my mum's name on it unlike my British passport and my Aus passports that didnt need my parents names on the cert).
Are you sure he had a long one showing his parents names ? As I find it really unusual that someone back in the 1920s would have had a long one, as if my family are anything to go by I would guess most people only ever had short ones way back then because of the cost and long ones weren't really needed for anything.
It's only in very recent years that some organisations may ask for the full ones. Don't know which ones though, as my hubby hasn't ever needed his long one for anything.
Unlike me, my hubby still hasn't got his long cert.
Kind regards
Ps I think a DNA test will be of some great help to you . ( you said you'll get your mum to test on the other thread). If you can't get her to test, your own will be the next best thing to your mum's.