How did Belmopan, Brasília, Abuja and Islamabad do it?

Well all of these are deliberate decisions to build a national capital. They overcame the bootstrap problem by being funded by a pre-existing national tax base.

dozens of new cities built just in Singapore during the past half century

Again, government funding is used to overcome the bootstrap problem. Singapore is also geographically small, and many of these "cities" would be characterized as neighborhoods if they were in the US.

Las Vegas

Well, wikipedia says it began life as a water resupply stop for steam trains, and then got lucky by being near a major government project - Hoover dam. Later it took advantage of regulatory differences. An eccentric billionaire seems to have played a key roll.

There seem to be several towns that exist because of regulatory differences, so this seems a factor to consider - at least one eccentric billionaire seems fairly serious about "seasteading" for this reason. Historically, religious and ideological differences have founded cites, if not nations, so this is one way to push through the bootstrap phase - Salt Lake City being a relatively modern example in the US. Masdar City - zero carbon, zero waste - is an interesting example - ironically funded by oil wealth.