SWIM was thinking that 1,4-butanediol could be formed by a grignard reaction with formaldehyde and 1,2-dibromoethane... would this conceivably work?

a3d0a3m

No, as you cannot form a grignard reagent of 1,2-dibromoethane.

Br-CH₂-CH₂-Br + Mg -> H₂C=CH₂ + MgBr₂

if you're ready to such lengths to make a simple
aliphatic alcohol, why not reduce succinic (or fumaric
or maleic) acid? just buy a bottle of vodka.