I always assumed people went for fino when the Adonis and the Bamboo were rediscovered because everyone thought this should be a dry drink and fino was the most easily, widely available dry sherry --certainly to bartenders. It’s also the style @RobertHess highlighted in the Trident so it was an ingredient of interest early on.
The only bartenders to have specified what type of sherry were to be used in their drinks are --unsurprisingly-- the Spaniards. Sanfeliu usually is quite good on this, Chicote less so. 99% of the time, they call for a dry style. Unfortunately, they don’t specify anything for the Adonis or the Bamboo (the latter features less often), maybe because those were drinks they encountered in other books and not out there in a bar somewhere.
As for what style of sherry was used originally in the Adonis, well… A friend of mine who has extensively worked with the Consejo Regulador (the regulating body) is adamant that the dry stuff didn’t make it to the US in any significant number at the end of the 19th century. Export sherries were mostly sweetened / off dry, he says. I do think, however, that @Splificator, doesn’t share that belief. He may even have data to share!