Yup, Weebly allows you edit HTML. Some (outdated) documentation showing the HTML editor is available here: http://themedocs.weebly.com/index.html.

Weebly's markup is good- especially compared to most website builders markup with poor markup (as you pointed out). Wix probably has some of the worst markup among website builders.

For reference, here's an example Wix website: http://s99433.wix.com/stevebenjamins and here's an example Weebly website: http://weeblyreviewed.weebly.com/. You can check out the markup using your console.