Gate needs a 'slam stopper'. Someone suggested a door stop at the bottom (trip hazard) - which would stop some of the force of closing.

What you need is angle or some vertical strip that the gate arrests closure against...and the only force the latch has to deal with is the impact of the bar itself, rather than arresting the momentum of the weight of the gate (slammed, wind, or just spring/weight closure)...it's a lot of leverage that gets applied over and over.

Once you deal with stopping the gate in other ways re-securing the latch is easy enough.

Edit: other solutions are:

Soft closer for the gate. They make hydraulic and spring closers that close the gate more slowly. This can also be as simple as a string connected to the gate and the opposite side with a weight in the middle.

Depending on aesthetics it could be as simple as a chunk of wood between wall and the latch bar. You'd need to use fasteners into better attachment points - but there is likely only 1 stud in there (which you've hit with the farther 2 screws presently). This horizontal wood would help with the problem...but not really 'solve' it...as the screws are resisting the force in tension - and screws work best in sheer. You'd also need a chunk the same thickness at the gate latch to keep the gate at the same closed position.

The surfacing is likely ~3/4-5/8" thick...a better latch bar with a longer attachment plate would help - but you still have the problem with screws needing to 'grab' something. There are better fasteners - such as metal 'screw in style' drywall auger type attachments. Also butterfly clips would be good too - except that at the end of the stucco - as you already know - nothing out there to grab on to. But always if the force is applied in tension you'll just delay the problem - which is why I suggested a slam stopper. Typically this would be a 1x1-1/2 angle with the short end sticking out and the screws penetrating deep enough to embed into the stud - the longer the angle and the more screws the more the energy of the gate closing would be spread out.

If you're willing to move the gate in farther (away from camera location) you could just have a latch coming straight out the wall.

But really...as this is inside - I think using 'standard' gate hardware isn't the right solution. I'd look into other forms of 'latching'...and I suspect the goal is to keep either kids or animals out of the area...does it need latching at all?