Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall
x
x
x
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x
Now let's look at the veritcal space next to the wall
yx
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yx
Finally we need to have the flat part on top of the wall, and the space above that
yyyyyyyyy
yxxxxxxxxx
yx
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yx
Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).
yyyyyyyyy
bxxxxxxxxx
yx
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ax
If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).
The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.
This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.
45' wall climbs (and lower) are safe though.
There are two different things happening here: your movement speed being halved, and spending half your movement speed. These work differently, and the order matters. The end result is that the grappler can't move-drag after standing up from prone. Here's how it works:
You start with your full movement speed.
Let's use 60′ for the sake of example.
Standing up costs half your movement.
Your current speed is 60′, half of which is 30′. To stand up you spend 30′ of movement. Your speed is still 60′.
Dragging while grappling halves your movement speed. Your movement speed this round has now been reduced to exactly how much you've already spent, so you will have 0 feet of movement to spend.
Your current speed is 60′, but attempting to move-drag a grappled opponent changes it to 30′. You have already spent 30′ of movement and have zero feet left to spend.
The end result is that after standing up, you still have half your movement left, but as soon as you try to drag a grappled opponent you will have no movement left and remain where you are, so realistically you won't bother trying to drag that round. (You could still end the grapple and move your remaining half movement though, of course.)
Best Answer
No
I have my own issues with the answers on the linked question, and have posted an opposing answer there, but I will rehash the point here.
The rules for movement on a grid state (PHB, p.192)
This does not say all movement distance (or required expenditure) is calculated in 5 feet increments, it only describes how the player expends their movement. Therefore, the cost of standing up while prone is still 2.5 feet of movement for this character. (Probably rounded down to 2).
If the player has 0 movement remaining, they do not have enough movement (2 feet) and therefore cannot stand up while prone.
TL;DR the premise of this question was based on (what I believe to be) faulty assumptions made within the answers of the parent question.