Why is it "borderline broken"? Are you referring to White's significant advantage after 1.Nc3 or else 1.e4, which could be easily resolved by playing Three Checks 960 instead?
I have no experience of Four Checks, are there sample games anywhere?
3-Check vs 4-Check

The difference is that with 3 check the game is quite violent and pieces are thrown at the king. 4 check allows for some strategic maneuvering without it being so violent. A mistake in 4 chess does not have to be fatal, while in 3 chess it can be a point of no return.
The difference is that with 3 check the game is quite violent and pieces are thrown at the king. 4 check allows for some strategic maneuvering without it being so violent. A mistake in 4 chess does not have to be fatal, while in 3 chess it can be a point of no return.
I haven't found my Three Checks games to be especially violent or lacking strategic manoeuvering, but perhaps that's because I've only played at correspondence speed, allowing plenty of time to plan and think. So maybe Four Checks makes sense if playing quick games? By the way, PyChess offers Five Checks games (but curiously not Four!).
I do assert that 3check is borderline broken, whereas 4check is not.