It looks like Black has a decisive advantage.
Obviously Black will avoid making any capture unless that gives her a decisive advantage normal-chess-wise. Meanwhile she can advance all her piece without fear of losing them.
White must be careful about his king safety: any open square in its vicinity is an invitation for a lone queen mate like ...Qf2# in OP's example, but surrounding it make it vulnerable to smothered mates (or decisive forks) by knights...
But his queen will also be very vulnerable to Black's minor pieces !
A simple plan for Black would be to develop everything on active squares, whether White control them or not: e5,d5,Nf6-e4,Nc6-d4,Bf5/g4,Bc5/g4,0-0,Q-somewhere, say Qf6 or Qh4, bring the rooks as well if need be. All without captures, or course.
And look for opportunities of any Q-check, N-check, B or N attacks against the wQ. I have no big advice for White beside keeping a tough formatiopn (but with Q-mobility!), occupying squares where the opponent wants to go, and hope for a mistake or a premature capture...
Playing several games with that rule (either between humans or engines) would help have a better feeling of the subtelties it creates, and probably settle the matter.
An idea to make the game less unbalanced (still advantageous for Black yet) would be to allow White to capture a checking piece, or even to allow all captures after the first check of the game is delivered.