1. Mother Gothel's plan is terrible
Disney villains never really think it through. Maybe that's why they're the ones who always end up falling into a bottomless pit or getting crushed by a boulder or getting sucked into a swirling green portal to hell. Tangled's Mother Gothel isn't much different, even though she's been around lots longer than Ursula or Frollo or that racist pug from Pocahontas.
If you remember, the movie opens with Flynn Rider magically narrating events he had no way of knowing transpired. As he tells it, a piece of sunlight dripped onto the earth, and on that spot a magical flower bloomed. This special plant had miraculous healing abilities, which would be great for the general public when it comes to sick children or this one mole on my arm that i'm not sure about -- but Mother Gothel hordes the flower power for herself.
Gothel's only illness in this case is the one everyone is afflicted with eventually: old age. So whenever she wants to get rid of the encroaching crow's feet around her eyes, she rolls on over to the flower, sings a special song and poof, she's youngish again.
While exact numbers vary from source to source, most agree that Gothel has been doing this for hundreds of years before Rapunzel's story begins. Hell, she's been suckling on that nectar since even before the kingdom was timelapsed into being.
So far, it's not a bad plan as far as evil schemes go. Even though Gothel has to keep renewing her youth more and more often as the years pass, in general the immortality racket has gone much better than say, Scar's questionable "Kill Mufasa and pillage the Pridelands so hard everyone is starving in five years" plan.
The problems start when the pregnant Queen of the nearby kingdom falls ill, and some soldiers nab the flower to heal her majesty. Not only does the Queen pull through, but the flower's mojo has been transferred to the newborn baby daughter, who we know as Rapunzel.
With the flower used up, Rapunzel's magic hair is the only thing that can restore Gothel's youth. And since cutting that hair renders its powers inert, Gothel naturally kidnaps the royal baby and hides her away in a tower. We see in the movie that Rapunzel's "mother" still very much depends on her "daughter's" hair to survive.
This was sort of a long walk to take to get to the gist, but that's kind of the point here. Gothel has been around a long, long time. At some point, she probably thought she'd live forever. But now that the flower is gone, all Gothel has is this girl -- a girl who will one day grow old, and die. And then where will she be?
Seriously, Gothel doesn't seem to be worried at all that her unlimited lifespan suddenly has an expiration date. She puts a ton of effort into appearing like a caring mother, all for a flimsy band-aid solution.
Now, this is pretty dark, but there is one way she could indefinitely increase her longevity: If Rapunzel had a child, the magic hair could theoretically pass onto the son or daughter. Gothel hasn't shown us anything to indicate that she'd be above breeding a flock of golden-haired life machines. Hell, if the family grew out enough, she could kidnap and imprison them all over the land, giving her convienent fill-up stations wherever she goes. And yet, when Rapunzel shows interest in a guy who could give her that life-extending child, Gothel brushes it aside.
That romance is her only shot at living into the next century, and she's still still overprotective. To be fair, that need to hoard power is part of her character, so it's only fitting that not sharing it would lead to her demise. Then again, you'd think that someone who's managed to survive for centuries would be a tad smarter than Gaston.
2. Why does Rapunzel know her birthday?
Gothel isn't just an idiot for not addressing the temporal problems of her royalty-kidnapping plan -- she's also the one responsible for inspiring Rapunzel to leave in the first place. Yes, after years and years of being cooped up, just about any teenager would probably want to leave the nest, but mommie dearest has done an exceptional job at keeping her prize posession locked away.
But Rapunzel doesn't leave just to leave -- she wants to go see the lights, the ones that float in the sky once a year. Rapunzel is so obsessed with these sparkling mysteries that she paints them on her wall. And hey, she's got a pretty good reasoning behind her fascination.
Imagine that you're trapped in a tower with nothing to do but clean, read one of three books or trade helpless shrugs with the local chameleon -- you'd latch onto any mystery about who you are or where you come from. And if the sky lit up on your birthday, like a million candles on the cake of the far horizon, you'd do anything to find out the truth.
The viewers at home know that the lights are in fact for Rapunzel. Each year, the King and Queen join the kingdom in setting off thousands of lanterns into the night, in hopes that they might guide their lost princess back to them.
Of course, Rapunzel eventually figures out that the candles are for her, but she never should have gotten that far. Gothel's dumbest, most baffling move in her (again, very long) lifetime was telling Rapunzel that her birthday was on the same day that the lanterns lit up the sky for the lost princess. Literally any other day than her real, actual birthday would have avoided the events of the entire movie, and saved Gothel one 70-foot deathtrip to the ground at the end of the movie. Bett yet? Rapunzel is a captive who knows nothing about the outside world -- just don't tell her about birthdays at all!
That just goes to show you: Never do anything nice for anyone. Especially anyone you've kidnapped.