The same people that went to shul at night on Shabbos went at night on Yom Tov. There are various reasons why there was no takana to say birchas me'ein sheva on Yom Tov.
According to the Tur/Shulchan Orach in O'Ch 487, we don't say me'ein sheva on Pesach because it is Leil Shimurim, a protected night, so we don't need to worry about the dangers in siman 268.
(This does not answer why we wouldn't say it on a different Yom Tov and in fact both my Siddur and Machzor state we only skip it Pesach that lands on a Friday night. When Succos lands on a Friday night, we do say it.)
The source of the blessing on Shabbos night is the Gemara in maseches Shabbos 24b which says we say it because of sakana, danger. Rashi explains that people came to shul Friday night (as opposed to weekdays) and we are afraid they didn't hurry to shul and will be left there alone after the tefila finishes, so we stretch out the tefila bitzibur.
Shearim Mitzuyanim Bihalacha there asks why the blessing wasn't instituted on Yom Tov. (Apparently he was of the opinion we never say it on Yom Tov, and his sources indicate this as well.)
He answers only on Shabbos where melacha is not allowed did people accept Shabbos early, but Yom Tov when it is allowed to perform some melacha, everyone came late, and whoever was really delayed, didn't come at all. He also quotes the machzor vitri who writes another understanding of sakana was because on Shabbos one may not carry a torch. This implies, says ShM'B, that we are not worried about Sakana because one can take a torch.
He also quotes the Sfas Emes who says they would only accept Shabbos early (leaving some people to come late) whereas they did not accept Yom Tov early.
(He does mention that Sfas Emes points to Tosafos in Taanis 2b s.v. HaOver quoting a Yerushalmi that not everyone came to shul at night on Yom Tov, however if you look there you'll see this was not mentioned as opposed to Friday night where everyone came, but rather the point was that more people ("everyone") would be there in the morning. I don't have the Sfas Emes handy so I can't tell you for sure what his point was.)
ShM'B also quotes the Chidushei HaRan who says that this was only instituted for Friday nights because Mazikim are more common on Friday nights and Tuesday nights. This leaves the question concerning Shavuos which may fall on Tuesday night, which he answers by saying it is not such a common occurrence, so they did not institute saying it.
