I also think that the balcony should be reviewed for its usefulness, two meters deep anyway. How high would the costs be including railing etc? If you provide more details about your project, it would be easier to assess where/how you could perhaps better allocate this money elsewhere.
Because of our south-southeast orientation with large window areas, I also had problems with the sun, but at the same time, I wanted it to be as open as possible. Not least for this reason, we installed an air conditioner, which is what I would do instead of the balcony; you will definitely use that, whereas the balcony only maybe at some point. We ourselves have a balcony with a view of a small lake and the vineyards, where we sit occasionally depending on the sun’s position, but that was more like a gimmick for the two of us. With children or without a lake, I wouldn’t have done that.
Our glass surfaces facing the sun are equipped with external blinds, which I consider the best solution. During the day the slats are tilted so the room stays bright but the sun doesn’t glare. People are completely different in this respect and even the same person experiences light/sun one day one way and the next day differently, often depending on their mood or what they are doing or not doing. That’s why I don’t like automation here either, because no smart automation could know what mood I am currently in or which side of the table I’m sitting on.
For example, we don’t have any roof overhang, windows reach up to the gallery, and thanks to the external blind options, we never find the sun to be a problem, on the contrary. When the sun moves around in the summer and sets lower in the evening, it shines into the house from the west and heats it up quite a bit, then we darken the blinds in the bedroom. My worry that the house would heat up extremely strongly because of the large windows (also why we have an air conditioner) was not confirmed and it’s really pleasant and would be tolerable even without air conditioning.
A key factor surely lies in the particularly good insulation (especially the roof) of the house. We even covered the terrace area in front of the kitchen with a 4m deep roofing, so no direct sun ever gets into the house there, which isn’t a problem due to the other windows in the open plan area.
In short - it would be too simplistic to answer here with yes or no, because there is considerable context to consider, especially the personal perception of the residents.
Maybe you’ll provide more info; if you need, I can also illustrate my explanations.