You're attacking with a weapon; make an attack roll. An attack roll is usually versus AC. Are there any other mechanics in the game that are resolved by attack roll versus a saving throw? i can't think of any right now. Inventing one for this one weapon is what I mean by "messy". A save versus a static DC is unsatisfying to me because it circumvents the rules for making attacks as they apply to the net, i.e. disadvantage unless unseen, etc. It also makes the net more of an effect you impose rather than a weapon, so conceptually weird for me. I could see it being a special attack that resolves with an opposed Dex check, but again this abandons the disadvantage of the attack rules and interacts oddly with the Strength check to escape the resulting restrained condition, IMO. My solution is to keep everything the same and add a modifier to the attack to counter the target's armor. AC already takes the target's Dex into account better, afaic, than a save or a check does.
Emphasis mine: I'm not quite sure what you mean by that phrasing (it could be "I roll an attack, you roll a save" or it could be "a save
instead of an attack roll"). Battlemaster maneuvers work precisely like the former; there are none precisely like the latter to my knowledge, but as noted, there is the Disarm thing, and there's also Grapple. Grapple is explicitly an Attack, albeit a "special" one, and functions in every way like an Attack (you can make a Grapple check as one of your attacks granted by the Extra Attack feature), except that it doesn't use an Attack Roll nor the target's AC, it uses an Ability Check--Strength(Athletics)--contested by the target's choice of Strength(Athletics) or Dexterity(Acrobatics). And there's some precedent for DCs involved in weapon-based save effects via the Weaponmaster's maneuver DC, although I will admit that that works as a rider on a successful attack rather than
directly forcing a save.
With all those examples of situations where Attacks, made with weapons, don't have to be cashed out as Attack Rolls though, I feel like there's nothing really unusual about treating a Net as a "weapon" that forces a save in place of dealing damage. It provides a clearly different effect from most "weapons," doesn't deal damage, and is even explicitly listed as having the "Special" property, which is described as, "A weapon with the Special property has
unusual rules governing its use, explained in the weapon's description (see 'Special Weapons' later in this section)." Emphasis mine: we already know that the net is going to have "unusual rules," so the game is already fully prepared for nets to do weird and wonky things that aren't like other weapons.
If it still bothers you that a special weapon with unusual rules doesn't work like other weapons, consider making the net a
tool proficiency rather than a
weapon proficiency. That liberates it from being shackled to the rules for how weapon attacks work, for one thing, and removes the need for the clunky "you can only make one attack using a net, even if you can make multiple attacks" rule. Just define it as an Action to use the net and you're golden--can't take the Attack and Use Net actions at the same time. Unless you blow an Action Surge for it as a Fighter, which seems like a sufficient investment on its own (and opens a precedent for the "net fighter" gladiator archetype).