RAW: The flier can escape the surface.
Depending on our flier's normal and flying speeds and the condition of the water (normal or difficult terrain), she'll be able to escape just fine, it's only a question of how long it takes to get out of being prone.
If she's lucky, she'll fall, land, and not be prone because she took no damage from the fall, whether by falling less than 10 feet, monk slow fall, having cat's grace, etc. In this case, she can immediately spend her whole movement as her flying speed dictates.
If she's less lucky, she'll fall and land prone. She doesn't have a swim speed, so she needs to fall back on walking speed at +1x cost. 'Standing up' from prone in this instance costs her entire walking speed. If she has more flying speed than she does walking speed, she can then spend the difference flying.
If she's least lucky (such as with your harpy and her worst day ever), she'll fall and land prone in difficult waters, where the movement cost is bumped up to +2x. Now, 'standing up' costs her whole walking speed and a half. But she can use the Dash action, leaving her with half her walking speed to spare. She can still get to flying immediately only if double her flying speed exceeds the 1.5x walking movement she spent on standing up.
All the relevant rules are in the Player's Handbook, pgs. 182-191.
Your harpy's worst day ever is only one round long.
For your harpy (speed 20 ft, fly 40 ft, MM p. 181) in sleet storm (20 ft tall, 40 ft radius cylinder, area forces Dex save or go prone and is difficult terrain, PHB p. 276), it's even money whether she falls prone due to the spell being only twice as high as the threshold for not taking falling damage, but at the beginning of her turn the spell might knock her prone again at the beginning of her turn anyway.
Assuming it does, the difficult terrain forces her to spend 30 feet of movement to stand up, which she has if she Dashes, leaving her with 80 feet of flying movement minus the 30 she spent on getting up.
50 feet of flying movement, even if you rule that the airspace is also difficult terrain (the spell only specifies the ground), is enough to leave the sleet storm through the top (and from the sides unless you're within 15 feet of the epicenter).
"But what about..."
You'll notice that none of this addresses any of your concerns about wet wings or being too slow to take off. That's because D&D isn't a physics simulator. This is just what the rules as written directly say about your flier's ability to get out of a water landing, and are not (and will never be) able to address every edge case. The space between is up to your DM.
The rules do suggest that a DM may ask for a Strength (Athletics) check in order to make any headway against rough waters, and per Rule Zero a DM could certainly apply the concerns you've listed to nullify a creature's ability to fly under certain conditions. Yet they could also reject it for game balance purposes. It's all up to them and how they want to run their game.