The biggest issue limiting most government consider and majorly the reason why bitcoins adoption has been slow is in my opinion the anonymity involved and the inability for government to directly be a party to your transaction.
If there were no such issues, gaining widespread adoption will not even be a topic for discussion as it would have happened already or almost done being adopted I think.
Can the governments of the world let bitcoin become fully in use in their respective countries without pushing for lesser anonymity and a way for them to monitor the system?
Of course, anonymity is what governments don't like most. Anonymity makes them lose control, and this is the most important tool for them to hold power.
The integration of the world of cryptocurrencies and the one regulated by governments is slowly happening and it will certainly increase. This is simply because companies that want to operate in an official, regulated market must register and from then on are subject to the laws of the country in which they operate.
However, some services or companies that will only operate on the cryptocurrency market will remain on the market and these will remain anonymous.
In answering your question directly, I see no possibility that any government would agree to make any activity completely anonymous.