The base game can barely support activities beyond combat in the first place, so it's kind of hard to expect that an encounter with Thargoids would have any meaningful content outside of shoot 'em up. Where's the framework required for something more substantial?
I'm hoping that the BGS will be reworked/extended into a war-based model, so that instead of systems being contested by factions, they will be contested by humans and Thargoids. The extent to which either side influences a system determines how likely you are to encounter Thargoid threats there. This influence would be military - blowing up enemy ships or bases - or logistic - completing missions for the various navies or supplying goods which are in short supply. The more of those things we do, the stronger our influence. This influence would be shown on the galaxy map (shades of green for us, red for them, for example) so we can easily see the battle-lines, and where our help - military or logistic - is needed. Those who want pew pew pew can patrol the front line; those who want some risk can do supply runs to the front line; those wanting a quiet life can operate further back. Certain commodities needed for the war effort can be manufactured (read: appear from thin air), mined or collected from planet surfaces deep in safe territory, and need to be traded from safe to relatively safe to dangerous systems, forming supply lines.
Not quite sure how exploration would fit, but then exploration has always been its own thing really as you tend to be too far away for much interaction. However, maybe spying on enemy movements, mapping the extent of their influence, or searching for rarer but more useful (read: influential) materials and bringing them to deep space supply hubs where they can be taken back to the bubble and eventually the front line.
Something like that would hopefully have some appeal for everyone.