The entire plot of Star Trek Insurrection (a Starfleet Admiral in some secret conspiracy with a race of bad plastic surgery people to secretly evacuate a race from an immortality planet to process their immortality particles) if an immortality potion hadn’t already been invented in an earlier movie. Wrath of Khan.
The Genesis Device turned out to be a crappy way of making new planets, but a great fountain of youth. Fire off a Genesis torpedo into an area with some debris or gasses, then shoot some torpedoes full of old or dead people into the area. Recover the torpedoes and you have your own immortality machine.
Not true immortality. Just a reset for a few decades, but that’s all the Briar Patch in Insurrection was offering and this could be repeated over and over again.
Without the Vulcan ability to back up and restore minds, reviving dead bodies and plugging memories into them wouldn’t work. But by TNG there was technology that could do that for humans out there. We saw it in use. It didn’t work too well plugged into a computer or even into Data, but shooting a body into the area, recovering it before it becomes a baby and uploading the memories might work.
Even if it doesn’t. There’s still a way to extend life by decades. Memories would be lost, if the de-aging process followed a biological pattern, but plenty of eighty year olds might accept losing forty years of memories. So if immortality was really on the agenda, it was available.
The only hitch is that everything involving Genesis was classified, but Starfleet had the information. If they wanted to use it, they didn’t need the Briar Patch.