Wow. While I've been cruising through baptism records with no luck for Adelaide, you have all been highly productive!
As to Thomas Broadhead: his birth date is given as 26 Apr 1881. There is a baptism for a Thomas Richard Broadhead which states a birth date of 27 Apr 1881. (This is the one in Aston, son of George and Catherine). You have already dismissed him, but one day out for a birthday is sooo close! Of course it can't be him, as he is still with wife Sarah in 1925 records.... but what a coincidence.
It's like this family doesn't want unravelled!
There are several William Hall's with the occupation of carter, but as I look at them, there is nothing to attach them to Amy.
My thoughts are that either Amy and a William Hall had a "rendevouz", or Adelaide just made up a father for her marriage registration.
The Smith family in general doesn't seem to register a lot of the births.... not helpful!
I shall resume trolling through baptisms, and hope you have it all solved when I return!
(Well done, Monica and Osprey!)