Not Check Variant

I think it would be interesting if the player could still move if the king was in check, but then if the king got captured, then the game would end. 

I don't know if it enough of a variant though...

I have been playing a variant that goes this way, there are no checks or stalemates so if the king gets blundered or in a zugzwang it gets captured. It makes a difference mostly because these games are about our mistakes, so even if you had a way to save the king, if you didn't then you lose. I think not leaving a check is actually illegal because it would simply be illogical and antigame maybe ? but games also work the other way

not having checks, stalemates, or rules that prevent you of being dumb sometimes makes the game more brutal somehow I think

this variant called Congo you can also move you king ( lion) into "check" so if you want to lose the game (or makes a terrible blunder) you may have your king captured mercylessly

but of course this is not the only difference between chess and congo.. thinking about what you said, maybe it would only make the game less... friendly ? to people in our rates or below... I don't think it would change anything for anyone who's up the 1600 maybe since they are not blundering the kiNG I Hope

also much more destructive on bullet games

There is something somewhat similar to this.

Its called Capture the Flag Chess

The definition of "variant" is very broad and a bit vague. They could be as similar to or removed from the original game as you like it, so long as there's....there's.... See what I mean? What is the standard of a chess game, anyways. Checkmate? The fact that there is a King that can't be left in check? The fact that there's a checkerboard? There are already "variants" that defy those. Apparently, some chess games don't require a King, and checkmate isn't even the end goal. I would think this defeats the whole purpose of chess, by definition. But, to these chess inventors, "chess" is more of a loose open book: as long as there's some checkerboard or a mere board that contains "slots" where game characters can land on and move across, their game is considered "chess". And, then, there are those very off-beat radical inventors who disregard even those 'requirements'.

(On the opposite end of the spectrum, my a larger, more complex variant, which could be considered  Monopoly-size, requires that chess pieces seek out and bear ring accessories around themselves (Sumrings, I call them in the game), before they can acquire abandoned rune pieces and turn those runes into new 'dangerously' powerful chess pieces to be added to a player's army. At least two people who learned of this variant heavily noted the new dimensionality of my game; that the seeking of runes and making new characters as the game is so removed from 'regular' chess, I should take out the chess base-component and make it a separate game altogether, rather than keeping it at as a chess variant.)

Chess variants come in all different levels of similarity and quantity/complexity of new rules.

With that being said, I think a checkmate-less game could, by the chess variant community at large, still be considered a variant: your chessmen can still capture and be captured. That's probably good enough for them. ^-^

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You should play a game with someone and post how it went; perhaps post the recorded game, too. 

In practise, this is the same as saying there is no stalemate, which has been suggested numerous times. Only that your suggestion allows someone to miss a check and immediately get punished for it, which is perfectly fine as well.

In older shogi variants, the winning condition is to capture the enemy king (and prince, if present). The king is allowed to move into attack, and you are allowed to capture a king that does so. It's a simple way of thinking, without having to consider the implications of "check". 

In chess, the main point of objection to an elimination of stalemate is that pawn up endgames would all be trivially won, which takes away from the aspect of a hard-fought draw.

Yes, sometimes I am in a position where I probably should resign, but most times I keep playing, and usually, I am stalemated, which is better than a loss in most cases.