I would say that staging matters, yes, even more nowadays.
The stage, however - whether it's an hi-tech orgy or a simple floor with black curtains in the background - does affect every act taking part in a same edition the same way: the way the stage is designed puts everyone on an equal footing, it is how a country is able to make the most of it regardless of its flaws that maximises their chances of success in the visual department.
Every year, we have examples of both hot visual messes and refined stagings (backgrounds, lights, props, camera works, etc.) and absolutely everything is possible on stage: one can set an intimate atmosphere in an arena or put on a show in a small venue. It is how a particular song is presented that matters imo.
Of course, that doesn't mean there aren't bad stages: for instance I thought the stage in Moscow was too big (most acts seemed lost in a LED ocean) while in Norway 2010, the lack of LED screens was a shame imo. BUT in those particular editions, there were still nations that made good use of what they had to work with.
If you ask me if any song would have had the same placing 10 years before or after, I couldn't tell because the results we have each year is the product of multiple factors, whether they are of a musical (compositions, lyrics, live performance, artist's charisma...), visual (staging, orchestra, camera work, hi-tech...), technical (number of participating countries, existence of jury vote and/or televote) or sociological nature (year the contest was held, evolving expectations of the viewers during the show, and the multiple reasons why someone would vote for a particular act...).
What I would say is that at the very beginning, Eurovision was purely about music since it looked more like an indoor music festival with no fancy distractions on stage. Now it has rather become an entertainment show where visuals can be so significant that they can help enhance not-so-strong songs musically-speaking. The contest has changed so much since its creation that I think it's pointless to assume what would have been the outcome in a different year.
If you just meant the stage at the exclusion of all the other factors (for instance, if we had had a stage like the 2001 one in 2016), then I'd say: most results would have roughly probably been the same.
