Net attacks RAW are made perpetually with disadvantage.
They act as a normal ranged attack, so you are either always throwing them at long range (10-15 feet) and incurring disadvantage, or you are throwing them at short range and as a result are in close combat and throwing them at disadvantage. The description of the net in the PHB gives it no special rules for avoiding either of these two general rules.
Mike Mearls confirms this in a tweet:
Do nets (thrown) use STR or DEX for attack rolls? Are they exempt from close combat disadvantage, as normal range is only 5feet? Dex, since you can't make melee attacks with it. still take disad in close combat despite range. -M
My only thought is the designers worried that nets might be overpowered and that a net attack should always be made at a disadvantage unless the PC is receiving advantage from somewhere. Nets actually seem really underpowered even without the disadvantage problem. Burn an action to restrain a creature who can escape it easily (DC10 or 5 slashing damage), but maybe they worried about the potential for a net dog pile where one player nets an enemy and then everyone makes their attacks against a netted enemy before that enemy gets a chance to escape. In that light it could be broken against single, powerful enemies.
Proceed with Caution, you are entering dangerous territory.
Expanding inspiration in this manner will make it more powerful (by definition). Adding dramatically more options to any ability will do this.
There are also some soft differences between granting success versus granting failure. All party members succeeding once will have a very different feel than a monster who fails four to six consecutive checks.
But these aren't the biggest threat you're going to face. That one belongs to the pure casters...
Save or Suck
While an attack roll on a spell versus a saving throw for its target often seems like an arbitrary distinction, they aren't always. There is a category of spells often referred to as "Save or Suck," which almost universally favor saving throws instead of attack rolls. Spells like Hold Person, Banishment, Feeblemind, Entangle, and Polymorph all inflict near-catastrophic effects on a failed saving throw.
Under your proposed house rule, an optimal tactic would be for players to transfer their inspiration to the full casters in the group (Wizard, Cleric, Druid, Sorcerer), who then uses it to cast repeated save or suck spells on any monster who stands above the crowd.
This has two very negative consequences:
Players are now under social pressure to transfer their inspiration elsewhere, rather than using it to help their own characters shine.
"Boss" fights become incredibly difficult to stage, as any boss has to deal with multiple saves against incapacitating effects with disadvantage applied.
This is not a fatal flaw...
This is not a fatal flaw to the house rule. Monsters written to be boss monsters typically have auto-save abilities, because even without disadvantage save or suck spells are pretty devastating.
In addition, exploiting it requires players to play at a somewhat optimized level. I would not be surprised for an individual group to either not see this tactic, or to be chivalrous about not exploiting it.
But be careful. Make sure that everyone involved is aware that you may back this change out early if it causes problems.
Best Answer
No, you would not have disadvantage
The relevant text about the oathbow is this:
Nothing in there indicates that you would have disadvantage with the bow against other targets, only when using other weapons (even if you use them against your sworn enemy).