(sorry for the necro)
 
Really, would Cozy Glow be a “true” alicorn? I’d be inclined to say “no,” because she wouldn’t necessarily have Earth Pony magic involved with this powerup, right? She was supercharged with Grogar’s magic, wasn’t born into, or “properly” ascended.
 
And in any case, unlike every other alicorn, when her power was drained by some means, the physical changes 100% went away.
 
Now, granted, we don’t know for sure if that would be the case for Flurry Heart (I’m inclined to say “she is a ‘true alicorn,’” though), on account of her magic never being drained, unlike the other four, who were still alicorns.

@Background Pony #AE81  
You’re right, she didn’t earn them, she was born with them thanks to her mother; Princess Celestia. 2 days before Nightmare Moons’ return, Celestia had experienced a prophecy foretelling how her then 4-year old daughter Cozy Glow Might bring about the end of Equestria, and Celestia acting purely on instinct instead of rational thought used the Elements of Harmony on her daughter to remove her horn and seal away her Magic, erasing everyponys memories of Cozy’s very existence before straight-up banishing her only child from Canterlot. It was only a day later when Celestia found out that it was all for nothing: she failed to account the possibility that the Prophecy was one of the SELF-FULFILLING ones, and instead of reaching out to find and apologize to her only child that she cruelly sent away, Celestia simply ignored it; hoping that Cozy would start a new life somewhere else (the same aloof attitude that she uses for any and all threats that she puts up with, explaining why there were no manhunts for the Changelings following A Canterlot Wedding, or why the guards are useless) and instead try to fill the void in her heart with Princess Luna and Twilight Sparkle.
 
Little did Celestia know was that just as the Elements had affected Cozy Glow, so too did they cast a curse upon Celestia that made it impossible for her to ever again defeat any threat to Equestria that she could have stopped: karma since it was because she wanted to protect Equestria and preserve her image of flawless perfection more than she wanted to care for her own family. This curse is why we have never seen Celestia win any of her conflicts directly on-screen, because she literally can never win again. The only way for her to get out of this is to come clean and tell the truth about what she did to Cozy Glow and to Luna…

@Inkan1969  
Oh yeah, my bad.

@Phoenixflambe  
Amore was never an Alicorn.

@CMWaters  
Ninth if counting Pinkie Pie or 10th counting Amore.

@Background Pony #6671  
No. It doesn’t. She never earned them.

@Background Pony #2E37  
Still counts.

Best alicorn did earn it. She was cunning.

@Background Pony #4122  
Earn it.  
 
Meh.

Enhancements made by Grogar’s bell magic don’t count. You gotta EARN it if you wanna be an alicorn.

Give Glimmy wings!

@CMWaters  
This just in the dream world.

7th, actually.  

Bruh