Note: This is not asking for a general strategy, you can find that here.
Is the following a valid strategy in solving Numberlink (Flow Free) puzzles (I like to call it 'unwrapping the grid'):
1...3 24.1. ...4. 3..2. .....
Since the 3s are on the edge of the box with nothing in the way, we can connect them:
1...3 24.1│ ...4│ 3..2│ └───┘
Then we can ignore the 3s:
1... 24.1 ...4 ..2
Now, the 2s and the 1s are on the edge of our remaining figure, also with nothing on the way – connect them:
1──┐ 24.1 └┐.4 └─2
Now, ignore them:
And the 4s are on the edges:
Now, put it all together to get the solution:
1──┐3 24┐1│ └┐└4│ 3└─2│ └───┘
In this case, it solved the whole puzzle. In other cases, it may be good to make progress.
But does it actually work?
Affirmative case: please provide a proof
Negative case:    please provide a valid counterexample
(noting that the base puzzle must also have a unique solution)
Edit: Jaap Scherphuis has provided a counterexample where there is a direction ambiguity. Is there a counterexample where there is no direction ambiguity (i.e. one of the boundaries has another number on it or there is only one path either way), because the strategy can't actually be used with such an ambiguity?