The phrase daddy long-legs covers at least three unrelated animals: harvestmen, which are arachnids but not spiders and have no venom glands at all; crane flies, which are insects with no venom; and pholcid cellar spiders, which are spiders and do have venom.
Only the last is even a candidate, and the claim collapses there too. Pholcus phalangioides has short fangs, but tests have shown it can pierce human skin, producing at most a brief mild sting. No study has found its venom to be unusually potent to vertebrates, and the story appears to have no identifiable source in the literature.
What the spider does have is an effective anti-predator move and a taste for other spiders. When threatened it vibrates its web so violently that its outline blurs, and it kills larger spiders by casting silk over them from beyond fang range.