“Something needs to break the log jam. Will it be Elon Musk or Jeff Bezos?”
Yes, the private sector.
I think that as the technology brings costs down, it will (this decade) pass the point where only governments can do spaceflight, and let rich people, corporations, or other large organizations (like a church or NGO), start taking off on their own for flights. After we have two or three permanent settlements elsewhere, there will be enough know how for all kinds of groups to strike out on their own to settle.
Cheaper lift is happening. New propulsion systems are a tough to forecast wildcard. Long term life support habitats is a challenge for the 2020s. Significant self-sufficiency will be achieved some time after that. Plentiful energy sources would speed things up - but nuclear could do a lot, and solar is getting better and could be significant (inside the asteroid belt where you can still get good sunlight).
The 2020’s will also be the start of a great explosion of robotic capability, which will really push power down to smaller groups with less money. With enough robotic capability, even individuals could head out on their own.
In general, once technologies are developed, the cycle time for copies to become much cheaper and widespread keeps getting faster - when supersmart robots are 3-D printing stuff, it will be quicker still.
So it is likely that the speed of space colonization will continue to accelerate rapidly, once the initial capability is developed. my guess is that we will have initial operating capability in the 2020’s, and be off to the races in the 2030s, with many groups and Nations establishing long-term facilities off-Earth.
Despite that hoopla, there is no engine made in the US which is as powerful, cheap, and reliable as the Russian RD series. To make matters worse, the Russians are working on new technology to make the Apollo engines look small - we have nothing,
As we buy our heavy lift now from the Russians, we will continue to do so in the future because they will remain cheaper, with more lift, unless US companies get a clue.
Musk and the rest will remain bit players working on so far non-profitable ventures paid for with tax payer subsidies. There is no vision there, just vague ideas unformed and undirected, searching for a grant or subsidy. Like a well run country, there needs to be leadership in space - as we displayed during the early part of the Apollo program before Nixon’s complete loss of political will set in - something from which the country has not recovered from. You will know a recovery is underway when education begins to return to the standards in place in the 60s and before.