Also, it's a misnomer that there is a single set of "big players." Wealth is a continuum. At every new level of "market cap" bitcoin reaches, there is a new set of "big players" it becomes interesting to.
Bitcoin is now at the level where it becomes an interesting investment for someone who wants to put a million dollars into it. It is still probably not at the level where players who need to invest $10M or more (medium-sized hedge funds), are really interested. But with another doubling or two, it will become accessible to them.
The biggest players, aside from central banks, are pension funds, who manage tens or hundreds of billions. We will not be seeing them for a long long time. They move as a herd, and they haven't even really begun to move into gold in any significant way (my guess is under 2% of pension assets are in gold), much less the teeny tiny asset that is bitcoin.
But I think it's counterproductive to focus too much on big players. It's all about the grass roots word of mouth, spreading the word as broadly as possible. Because of the network effects of bitcoin, it is much more valuable to have 100 people who want to buy $10k worth than to have 1 person who wants to buy $1M worth. Metcalfe's law says the value of a network is proportional to the square of the number of connected nodes. That means that every new person that becomes a holder/user of bitcoin is more valuable to the ecosystem than the previous user. Not only that, a broader base of owners is much more stable. The person with a single large holding might suddenly decide to sell it all. All 100 smaller holders are very statistically unlikely to do so.