Most definitely not. One of the main selling points of bitcoin is to have a currency that's completely public. If Bitcoin will only be controlled by a chosen small number of people, what's would be the point of it?
Exactly, also, if Bitcoin was considered a private cryptocurrency, the amount of exposure it'll get is pretty low compared to what is happening nowadays wherein influencers, celebrities kept on bragging about how they want to invest and make profit out of Bitcoin, it's a completely different game if Bitcoin was only accessed by a few authorized people, it may still go on if it was a private currency, not just huge and #1 as it is today and definitely a few people would know the word "Bitcoin".