gdp growth =/= energy consumption growth
Also 1.03^1000 = 6.810^12, but wolframalpha says the number of stars in the milky way is 310^11. Considering that we're nowhere close to capturing even 1% of the energy output of energy that reaches the earth, let alone all the energy that the sun emits, your estimate of 1000 years is probably off by a few orders of magnitude. Finally, if we're actually capturing all the energy of the milky way, presumably we'd be a space faring civilization and can therefore colonize other galaxies, making that a non-issue.
As far as data is concerned, the correlation is pretty big though: https://theshiftproject.org/wp-content/uploads/2020/05/gdp_e...
It's interesting to see how world trends look as more and more of the world ends up in the position the US was in 20 years ago in terms of economic mix.
That argument is built on the fact that we don't find efficiencies. A lot of economic growth is simply much more efficient use of resources, ie less resources and much greater return. A couple of decades ago you needed a lot of expensive copper to connect a city with phone and slow internet access. Now all you need is cheap thin plastic for gigabit+ speeds.
A computer 70 years ago was an enormous contraption made of of literally tons of expensive metal components, while a raspberry pi zero has a tiny bit of copper, resin and silicon. The latter is vastly more powerful and cheaper.
Sure, growth can't last forever, but we have a long way to go.
If we continue growing the economy at 2-3% YoY we will be extracting all energy from the Milky Way in 1000 years and applying it to the economy. Not probable!
Clearly there is some transition to the upper part of this S-curve and things need to change when we get there. I wouldn't be surprised if we already are there for most sectors.