Reading again through the other post, just wanted to add again here:
Could the records have been destroyed in a fire?
I have been informed by a cousin that in 1944 Thomas needed his birth certificate to complete a grant application for my aunt to attend grammar school. They travelled to Scotland (Dumfries?) to the records office and were informed that some records from that time, including his, had been badly damaged/destroyed in a fire. (He was issued with some documentation for the grant to be processed).
Have seen this kind of family lore come up before on a number of searches. All the statutory BMD records for the GROS are maintained in Edinburgh from the start of official registration in 1855. Copies of registers were inspected/copied/sent to Ediburgh by all registrars. Any possible fire would have been at a local level, but records are also in Edinburgh. Given that you have been using SP, the online window to the GROS now, and cannot find anything so far connecting Thomas to a father George would make me think that the family stories are not what they seem....but you likely know that already for sure
Have you considered following through on the possible school records you refer to? Many schools have kept great historical records of pupils.
Monica