I'm going to answer with anecdotal evidence from Finnish people I know, not first hand.
- Is the labor shortage real, or is the available labor simply too expensive for local tastes?
Both, but mostly the first. A friend of mine chose to work for a local consulting company because it paid more than certain well-known Game Engine company in USA. Furthermore, the Finnish company was more "agile" in the recruitment process, meaning they probably weren't feeling picky, but rather slightly desperate. Caveats: it's just one case.
- The Finnish education system was held up as the most successful in the world when that was a topic a couple years ago; why can't it produce more tech talent?
There seems to be a mismatch between how much Nordic people are in need of the money and job security that a tech profession brings, and the industry needs. In other words, with a social security net, more people pursues other careers, even if they risk finding themselves jobless at certain point.
- Does the country experience a significant brain drain of (English-speaking) local engineers?
Hard to answer that one, but it seems the opposite. Last time I checked, Finland had a downright friendly immigration system for qualified people.
- Given the circumstances, why not try to use more remote labor (e.g. in EU countries, within the same time zone) rather than trying to move people to Helsinki?
Companies do use remote labor, but I don't think that's a policy the major of Helsinki would go out of his way to promote publicly. Also, when it comes to tech talent, the EU has a peculiar situation: some countries have affordable tech talent slightly in excess, and some countries are searching for affordable tech talent. So there is market competition for the same talent. Hiring out-of-border is not an automatic guaranty of meeting needs.
There is another factor, and that's that Finland invests heavily in social security. A local engineer benefits from those investments just by living in Finland. For example, child-care and free, high quality healthcare. Add other benefits the employer already gives to the local workforce, like access to a gym, local cafeteria, and work facilities. An employer may believe they can extract higher-quality labor from somebody by relocating them to a place where they enjoy those investments, and that would come practically at no extra cost for that employer.