I had to go back through the episode again. But you are right the dad is wearing the necklace. (I was confused because earlier in the episode he’s wearing armor.)
 
As for why in the movie why Novo doesn’t turn them into hippogriffs after kicking them out of seaquestria. She’s trying to punish them and make them ‘go away’. Besides they were transformed for the length of one song. It’d be cruel to turn someone forever into a hippogriff when their natural form is a pony. Because remember in the movie she is using the pearl like a magic wand to cast magic at them. They would be stuck with whatever form she decides to give them, not switching back and forth.
 
(Alternatively it could be what happens when she withdraws the transforming magic that they fall back to their ‘natural’ state)
 
The thing about the hoof flipper / pointed ear thing. I don’t think it determines the direction of being hippogriff vs seapony. It seems more about a Mount Aris native vs Transformed pony thing.
 
I know you want to say flippers=talons and backfins=wings. But the magic seems very variable. Like Sweetie Belle and Rarity get horns as seaponies and ponies have their cutie marks.
 
If Silverstream was born a seapony why does she have them? You by your own admission say that the rules are inconsistent. So maybe not get so wound up on it being ironclad in other ways? At the end of the day we are splitting hairs over something that I bet the writers of the show themselves probably didn’t think too deeply about.
 
Noone explicitly says it doesn’t work this way and nothing on screen shows that it can’t. So I’ll step into the empty space with my fanon. My Belle stole found her necklace. So maybe it’s still imprinted with the hippogriffy traits of the previous owner. See? Easy to come up with reasons to make things work.

@Fylifa  
If she found a hippogriff necklace and she was born a unicorn, it would only allow her to transform into a seapony and back into a unicorn if she was born a unicorn. Silverstream was born a seapony, sure, but she turns into a hippogriff when she goes to land because her father (and maybe her mother’s parents or her mother) spent their lives as Hippogriffs on Mount Aris before they were forced to go into hiding in Seaquestria. I find that unthinkable that a unicorn who turns into a seapony can turn into a Hippogriff if she were to go on land. My proof would be the scene where Queen Novo banishes the Mane 6 after Twilight’s attempt to steal the pearl. They were not turned into Hippogriffs after that. They were turned into their regular species. Also, there is a distinction between seaponies who were hippogriffs originally and seaponies who were ponies turned to seaponies. It is in the ears and in the flippers. When hippogriffs become seaponies, their talons turn into flipper rays. With the exception of Terramar (for reasons I am unsure about), when hippogriffs turn into seaponies, their ears also develop flipper rays.
 

 

 
PS: Sky Beak (father of Silverstream and Terramar) clearly wears a necklace too.
 

 
My hypotheses on why Ocean Flow (Silverstream and Terramar’s mom) does not wear the necklace is because either, she was born a seapony or hippogriff but either lost or refused to get a necklace after the Storm King was defeated and the hippogriffs were allowed to return to Mount Aris (maybe because she did not want to return). Or, maybe she has one, but she does not wear it.

@EthanRosca  
Not quite!
 
The hippogriff <-> seapony transformation is not a biological thing. From the show’s lore, Queen Novo broke up pieces of her transformation pearl from the movie and distributed it so they all have the shapeshift ability.
 
Silverstream in her first appearance in S8e1 School Daze mentions that she’s never seen the surface. IE she was born a seapony and later given the ability to go seapony -> hippogriff. That is to say she was never a hippogriff before and was born underwater.
 
In the Surf and/or Turf episode, the whole conflict is that Terramar’s mom and dad have different views on being seapony vs hippogriff. Neither seapony mom or hippogriff dad wore the necklace because they don’t want to transform to the other form. So you can be a seapony or a hippogriff without wearing a transformation necklace too.
 
We also see in that episode that they can share the transformation power to regular from holding hooves pretty easily. Like with scootaloo and sweetie belle
 
I am headcannoning a little, but I don’t see any reason why if a unicorn can turn into a seapony and back (Like Sweetie Belle) and a natural born seapony like Silverstream can turn into a hippogriff and back. Why it wouldn’t work to make a unicorn (who was never a hippogriff) into a hippogriff and back.

From what I know, the necklace only allows Hippogriffs to transform into Seaponies. How can she even turn into a Hippogriff? She was born a unicorn and Hippogriffs are a distinct species. The only species I know in the MLP Canon that can shapeshift are changelings.

@Fylifa  
unfortunately I possessed nowhere near the amount of money needed to do those kinds of options in fact I barely have enough money to pay someone else for something I owe them so unfortunately I can’t do one of those ych things I’m sorry

@Draco-knight-Blaze
 
The artist was very kind to work with me on adjusting their usual YCH offer to add in the extra details. They often bring these kind of sheets up for auction on that YCH website. You can visit their page here

This is a very well-made ref sheet it has the pony from all angles that’s how tall they are are compared to another Pony in Equestria as well as their form in seapony and hippogriff as well as there teen / Philly forum I like these ref sheets they’re very well detailed

Allow me just to say,  
“Oui, tu es belle!” ^^