The Action provided by Witch Bolt is not "Recasting" the spell
The first cast allows me to recast it until the target exits the spell range, moves into full cover, or I cast something else.
Emphasis mine, is where you're confused. No part of Witch Bolt involves a "recasting" of the spell.
Let's reread the description for Witch Bolt:
A beam of crackling, blue energy lances out toward a creature within range, forming a sustained arc of lightning between you and the target. Make a ranged spell attack against that creature. On a hit, the target takes 1d12 lightning damage, and on each of your turns for the duration, you can use your action to deal 1d12 lightning damage to the target automatically. The spell ends if you use your action to do anything else. The spell also ends if the target is ever outside the spell's range or if it has total cover from you.
Also emphasis mine.
What Witch Bolt allows you to do is use your action repeatedly on subsequent turns to reapply the damage, without needing to make another Attack Roll or force another Saving Throw. So long as the target remains in range, you have a guaranteed damage source for the duration. That's the strength of this spell. Nothing in the description says anything about "recasting" the spell; this is simply an Action used to invoke this damage. The only time you'll cast the spell again is if the duration ends (naturally or early) and you decide to cast the spell again.