Rainbow Dash: So how serious is this Princess power? Like, can we arrest anyone we don't like now?
Fluttershy: Maaaybe going full tyrant would be a bad idea...
Applejack: Doesn't seem like Twilight's style, anyways.
Twilight Sparkle: Still, you bring up a good point... This is a LOT. How much am I going to be expected to rule? Will I do it wisely? Will I be popular, or will I have to deal with those that hate me? Not to mention Discord...
DM: Celestia gently places a hoof on your shoulder.
Princess Celestia: Being an alicorn Princess means ponies will look to you for guidance, leadership, and protection. But like you said, all of that can wait until after Discord. And even then, we'll start slow. I'm sure you and your friends will be able to find a few perks along the way. Chances are, you won't have to worry about day-to-day expenses ever again.
Pinkie Pie: "Wait, I can expense my party supplies?!" Pinkie Pie does a spit-take!
DM: ... Thank you thank you *thank you* for not acting that out in real life.
Pinkie Pie: Well, yeah! There's too much paper in the blast zone!
The characters in the show generally made sure not to lean too hard on Princess Twilight's new power and connections early on (save for the Cutie Mark Crusaders). These are tabletop players, though, so just as a matter of practicality, they're going to want to squeeze as much of an advantage as they can out of this.
The mayor's still there, and Twilight never seems to have a formal leadership role. The very fact that most of her speeches are to announce danger would probably make her something of an albatross, and that was commented on in Episode 100. "Here comes Princess Twilight - what disaster is coming after us today?" "Doesn't matter, she and the Mane Six will handle it."
Indeed, Twilight never appeared to run the administrative parts of the town. Instead her castle converted to a school and she took over the administrative duties at the capital, so Ponyville never stopped being a democratic town.
Twilight is an Equestrian princess, not specifically managing Ponyville. In the US, it'd be like having the Vice President reside in a town: the VP doesn't directly make the town's laws or policies.
Right. Even modern royal dukes, while they derive income from the duchy, don't have any command role. The Duke of Cambridge doesn't actually get to tell Cambridge what to do.
Even the senior princesses seem to rule with a pretty light hoof. They don't interfere with ponies who know what they're doing. For that matter, in this comic Celestia can't get ponies tickets to her own gala.
Is she democratically elected? I don't recall that ever being mentioned or shown in the show
I always thought it was an appointed position, and the title of "mayor" was because of reasons (like how Celestia and Luna are only "princesses" and not "queens")
I noticed that, particularly in the bit where they were in Manehattan and Twilight couldn't get a taxi. No one even noticed the presence of a small alicorn. It makes me wonder if small, unassuming alicorns are more common than they appear - Cadance was able to be a babysitter, and wander around town without everyone dropping what they were doing the way they would if it was Celestia. No escort, either; stories about the children of presidents are common enough that the vagaries of security following you everywhere would make for useful story fodder, but nothing.
On the other hand, Pinkie can get that +5 party cannon now.
Twilight might not get an escort, but Flurry Heart may have a complex about her security detail in her adolescence.
I suspect that Celestia and Luna stand out by their size first, alicornness second. It's possible not to notice that an otherwise-ordinary-looking pony has both wings and a horn, but a literal big pony around town is harder to miss.
Yeah, if Rainbow Dash had a mane and tail like that, she'd be getting more attention - to the point that ponies would notice if she had a horn in addition to her wings. If she did, and Celestia-like size to boot, ponies would be scrambling to help with whatever she wanted done. (To her chagrin, until she got used to it.)
It's possible not to notice ... if the pony in question hasn't just been the center of a widely advertised and -attended coronation. Especially in Manehattan, where you'd think there'd be her cutie mark slapped on a skyscraper somewhere.
Except in this comic, where apparently half the population has saved Equestria at some point.
In the end CelestiaMod evaded the question of what happened to Discord if they are in Ponyville and how are they there... Maybe is better to not point plot holes
There's a story on FiMFic where it's a minor point that Celestia has given Pinkie access to the "Royal-Party Emergency Fund" account. Celestia still considers it a useful way of harnessing Pinkie Sense, in that if Pinkie senses some danger and needs money to head it off then it's right there. Naturally, Pinkie tends to misplace the hyphen, using the account for "party-emergencies."