You can't use those transformers at high freq, this generally requires custom-wound jobs. There are a lot of "critical" factors here like wire gauge, core type/size, transistor drivers, board layout, and switching freq that require careful consideration. Skin effect is a limiting factor.
So you end up losing the problem of requiring a huge transformer core and a crazy amount of copper wire, but at the same time trade it off for a whole slew of new design issues.
Larger supplies sometimes use multiple smaller transformers rather than one large one.
Note the discrepancy between "peak" and "continuous" ratings in these inverters. The output is often exaggerated.
Honestly there's no advantage to the idea of building an inverter nowadays. It would be incredibly expensive and probably wouldn't perform for crap, not the first few tries anyways. If you don't know what you're doing you could blow it up dozens of times. There's a lot of reasons to learn how to build switching power supplies though.