FUN token doesn't really react to that
The token price will react as BTC is the denominator which means when BTC is stronger especially sharply up on a particular day that should put pressure on that price. Same as prices are certain to rise when a currency value falls. So thats a definite relationship but I agree if people are positive on BTC it probably does help the FUN token price longer term. If we compare the recent low for FUN price in dollars and cents to the price today, its risen by 44% which is no small amount. To then want a Satoshi quoted price rising on top of that value gain would be exceptional, longer term I would estimate Bitcoin performing well raises the interest and requirement for 'safe harbor' of long term sites like this. Whatever the narrative I do think its a fairly good guess this site does well alongside bitcoin usage and market generally. However the price wont move immediately in parallel to BTC.
You are right, and I agree with all you said, but just to add that, a coin or an altcoin trading against bitcoin, can only experience increase in its dollar price when the price of bitcoin is going up, only if the bitcoin price of the altcoin remains the same or increase a bit, if as the price of bitcoin is going up, and the bitcoin price is the altcoin is dropping bit by bit, we can also realize that, even as the price of bitcoin is going up, the dollar price of that altcoin could remain exactly the same or might even be dropping depending how sharp or slow the bitcoin price of the altcoin moves down..
This is to say that, there are many factors working and contributing to the price of a coin, outside bitcoin's movement, even though bitcoin is still very much dominant in the entire market for now.