>>In English you can't say 'That he remembered her birthday made her happy', for instance.
Well, you can, actually, though many people no longer do, and almost no one would in such a colloquial context. But MikeE's example
>>That he did it is not in dispute
certainly works, and it's essentially the same construction. Indeed, style writers who care about this issue prefer it over adding 'the fact,' which in many contexts is just two extraneous words. The compromise solution that works in all registers is of course just to start with 'it':
It made her happy that he remembered her birthday.
It is not in dispute that he did it.
MikeE, I'm afraid your two sentences starting with 'Because' sound not just awkward but wrong, at least to me. The only way to make them acceptable would be to punctuate the 'because' clause as a quotation, in quotation marks, but to me that would still be awkward.
I think part of the problem may be that the sentences contain the noun 'reason.' I would also not write 'The reason was *because,' because to me that sounds doppelt gemoppelt, just as Steff says.
At least some style writers agree. Garner says 'This construction is loose, because reason implies because and vice versa. ... The best cure ... is to replace because with that ... Variations such as reason is due to are no better' (p. 558). However, he's okay with 'the reason why,' saying it's as idiomatic as 'time when' and 'place where.'
Burchfield (Fowler 3rd ed.) says, in the entry under reason (p. 655), 'Various studies of the construction the reason ... (is) because leave no doubt about its frequency ... In the 20c. the construction appears ... all over the English-speaking world ... If you look for the construction in your morning newspaper or in the unguarded pages of popular magazines and light fiction, you will undoubtedly find it sooner or later. But, for the present at any rate, its absence from the works of our most talented writers and scholars is more significant than its presence in more informal printed work.' In the entry under because (p. 100), he says it 'aches with redundancy' and 'is still as inadmissible in standard English as it was when H. W. Fowler objected to it in 1926.'
Burchfield also mentions 'Because (or just because) at the head of a dependent clause governing a main clause' as another type in the general group of 'More questionable because-clauses, with varying degrees of questionableness,' though his reason, that they 'demand too long an attention span before the onset of the main clause,' seems spurious to me. I think Steff's logical analysis is better. (-: