Seems like sun-mining would make a possible end-game source of resources, allowing you to afford some of the monster ships you can now design and build with end-tech,and provide more of an incentive in the early / mid game to take and hold territory. Even a system with no other bodies could one day be sun-mined, so it has inherent material value.
It might be wise to limit quantities and accessibilty though. Stars are vastly larger than any planet we could exploit, so we'd literally only be able to scratch the surface. Maybe not even the surface, maybe you'd "mine" the corona. Make it a supertech after Sorium mining perhaps, with a few levels for mining rate, like other extraction techs. Since you could tweak the costs and mining rates to suit, you could predetermine the expected breakeven date for the mining modules, based on an average star, and you can fiddle the distribution so "average" can fall where you like, and you could rig the numbers for mining modules to require lots of wealth / crew to limit their deployment too, if you like.
I don't think you need to worry about changing the behaviour of stars within the timescale of Aurora though. Even Pop 0 stars have a life of 10 mil years or so, and it's unlikely they would be worth mining anyway. Extracting millions of tons of anything is inconsequential to a star; consider, our own quite weedy star loses around a billion tons kilograms of mass per second in ordinary solar wind emissions. Any player mining, at any conceivable rate with any conceivable tech just won't even show up.
<edit> I was 3 orders of magnitude out. Still doesn't matter </edit>
