Interactivity is not about "you can play on your opponent's turn". It is important to understand that very few games, even not Magic, allow you to actually play a card while it's your opponent's moment to play.
Rather, they break the game into steps, where first you play, then your opponent, then you again. What Magic calls a "turn" is a much larger construct within the game that sort of decides who is the primary actor. Within the "turn" each player gets a whole bunch of opportunities to play, and the majority of game-objects are in the game for a while, first in "card" form, then in "spell" form, then in "permanent" form.
The feeling of interactivity comes from having the option to target cards in any of those forms, and being given a moment to play something almost constantly.
Really, what interactivity means is that any part of the game can have a meaningful effect on any other part of the game. Having the option for a card in hand to affect a creature, a creature to affect a spell on the stack, the stack to have an effect on the graveyard, a library affecting the player, etc. is what it makes it interactive.
So really what you are looking for is not "having a stack", but "having various different things that can interact and react with various other different things". A highly complex structure for handling "whose moment it is to play" is one way of doing that, but just having fairly short turns and lots of things that could influence lots of other things, in lots of ways, also works.
For a concrete example of a highly interactive way of doing a mage-duel, imagine the following. Each player is a mage. Each mage is busy casting spells, which are built from generic spell components (which are cards). When a spell is completely built, the spell is cast and affects the opponent. Until then; it's just sitting there, incomplete, ready to be interacted with. But the interactivity is that players can add spell-components to both players spells-in-progress.
So I could be building a single spell by dropping down the cards "Deal 5 damage", "Deal 3 damage", "Element: Fire", "Target discards one card", only for my opponent to suddenly drop a card onto that stack saying "Target: self" to completely change what is going to happen.
Or maybe my opponent responds to me building a Fire-damage spell by creating a spell that has "Shield 10 damage", and I quickly drop an "Element: Lightning" spell on that stack so that it won't work against my spell.
You can have tons of interactions between the players based on the spells they are building, and you don't need to copy anything from Magic at all. There is no "stack", spells are just on the board in parts. Turns are short (each player plays a card or launches a completed spell) and you can't interrupt anyone during their turn; but good spells take a few turns to build so you have plenty of time to mess around.