Ironically, at independence Kenya was at par with these countries. Though I cannot confirm the veracity of this, I have also on many occasions heard that some of these countries did borrow planning materials from Kenya - for their civil service.
Economies with strategically located container shipping ports developed via protectionist trade policies.
These economies were 'allowed' to develop into rich industrial economies was because America provided privileged access to the US domestic markets in spite of their protectionist policies.
They allowed that because they saw the value of having these countries developed into 'westernized' bulwarks against the USSR's sphere of influence.
They could all easily have remained poor under different circumstances.
It's a lot more free than it was until late 1970's and China has got tremendously richer as a consequence.
Nor is South Korea a model of freedom; it was authoritarian, it was and still is protectionist. But it did integrate to world economy. Compare it to North Korea, which started off as equally poor, and which has real, concrete, deadly famines in recent memory.
Same applies to R.O.C. Taiwan; autocratic, even corrupt, but reached out to world market (and as a consequence forced the mainland China to follow).
Singapore and Hong Kong: very free trade all the time, and way ahead of others in the region.
All of those countries enacted protectionist trade policies. Singapore/HK tended not to target specific industries but they suppressed the value of their currencies by loading up on US treasuries (their small size made this easy).
In all of those cases those countries were given tacit approval by the US to enact protectionist trading policies and yet maintain their access to US markets.
Protectionism and freedom are not absolute. Yes, those countries as well as the United States also engages in lots of protectionist policies.
However, the mentioned countries integrated more to world trade then most other countries in their region, and as a result they are now among the most developed.
Compare South Korea and North Korea, compare Taiwan R.O.C. to P.R.China prior to impacts of Deng's policy change, compare China and India.