But what happens if Bitcoin becomes way more successful than it already is?
Maybe you should also ask this question about gold because it has been a valuable and successful asset for a very long time without losing its value. Despite the success and economic value of gold, did it replace the Fiat currency? It didn't. The wrong idea that people keeps having in their mind is that there will come a time when Bitcoin will replace Fiats but it won't, if you are thinking towards that direction you should erase it from you mind because it's not going to happen.
What happens when millions or billions of people actually depend on it for their day in and day out activities?
If millions and billions of people adopts and use Bitcoin as everyday money of course it will cause more scarcity and the value increases.
Who or what then would control the infrastructure of such a massive market?
Bitcoin is not controlled by anyone or any government so if there is massive inflows it will only help to strengthen the bitcoin network.
What happens to Bitcoin's decentralisation when big institutions become major holders?
Big institutions will of course be major holders of bitcoin but did they or will they interfere in your own transactions? No they won't, so it remains decentralised.
Can Bitcoin remain the same when it becomes a global financial system?
Remain the same in which aspect? please explain.