I've said this before: weight is important. But it's not the only thing to think about. It's a case of matching the lift potential of the airframe to an acceptable weight envelope. If the plane's wing doesn't provide much lift due to say, airfoil, flap configuration or a number of other elements, then it can't weigh much or the wing won't be able to carry the payload adequately and that's when you get stalls in corners and the thing is falling out of the air at the most inconvenient times. If the lift potential of the wing is really big, then it can (and in some cases probably should) carry more payload.
Obviously, you can overpower the problem to an extent if it's close. If the plane is just a bit past the optimum (or even maximum) wingload for the configuration, you can use more horsepower to pull it through. It will probably never be a great plane, but you can sort of overpower it into one you can get away with.
Every design has an optimum weight. A lot lighter than the optimum, and you create problems with control response and will need to dial out a lot of control moments (among other issues). Too heavy and you go past the airframe's ability to carry the payload and you get all those wonderful problems associated with toads. Certainly control response is compromised and the stall speed goes up, sometimes dramatically.
Some designs, that optimum weight range is pretty narrow. But for most designs, and certainly most of the more famous designs, it's fairly wide. For instances, I'm told that the ideal weight for an Impact is 60-62 oz. Build it straight, at that weight and any one of a number of engines can power it successfully. I've seen Paul power one with everything from a OS40VF to and ST60 to a Big Bore PA to a 4 Strokes to now the equivalent of a 90. All flew the plane very well. But it's balancing the lift potential and design to the optimum weight is what makes a great flying plane. As long at the power delivery is consistent, the engine provides enough power to fly the plane well and is of the sort of run you plan to use (and you've trimmed for), I think the power is a secondary issue. But those are big ifs. I've certainly spent enough time trying to get all those ducks flying in formation.
Point of all this is, it's not one thing. I don't think you can make a statement like "lighter is always better" for instance (though many do). It's true that lighter is usually better, but if you have a really light plane that folds up in the first hard corner, or it isn't straight or what ever, it's still a toad. And when you drop off the bottom of the weight envelope, you start to have problems.
Just my view on it. I'm sure others will have different opinions. Certainly Sparky and I have argued this a time or two. But hey, that's what makes this fun.