My comments on these assertions:
We can't directly evaluate the posterior as the normalising constant is too hard to calculate for interesting problems. Instead we sample from it.
No, the normalising constant$$\in_\Theta \pi(\theta)f(x|\theta)\,\text d\theta$$being unknown is not the issue for being unable to handle inference from the posterior distribution. The complexity of the posterior density is the primary reason for running simulations. (The normalising constant is mostly useful to compute the evidence in Bayesian hypothesis testing.)
We do this by engineering a Markov chain that has the same stationary
distribution as the target distribution (the posterior in our case)
This is correct (if one possibility). Note that MCMC is a general simulation method that is not restricted to Bayesian computation.
When we have reached this stationary state we continue to run the
Markov chain and sample from it to build up our empirical distribution of the posterior
Not exactly as "reaching stationarity" is most often impossible to detect/assert in practice. Some techniques exist, but they are not exact and mileage [varies][5]. Exact (or perfect) sampling is restricted to some ordered settings and very costly. However, the ergodic theorem validates the use of Monte Carlo averages in this setting without "waiting" for stationarity.
All Markov chains are completely described by their transition
probabilities.
The generic term is transition kernel, as the target distribution often is absolutely continuous. Some MCMC methods use continuous time processes, in which case there is no transition kernel stricto sensus.
We therefore control/engineer the Markov chain by controlling the transition probabilities. All MCMC algorithms work from this principle but the exact method for generating these transition probabilities differs between algorithms.
Markov chain Monte Carlo algorithm are indeed validated by the fact that their transition kernel ensures stationarity for the target
distribution$$\pi(\theta'|x) = \int_\Theta \pi(\theta|x)K(\theta,\theta')\,\text d\theta\tag{1}$$
If we have a particular algorithm for generating these transition
probabilities, we can verify that it converges to the stationary
distribution by using the detailed balance equation on the proposed
transition probabilities
No, detailed balance is not a necessary condition for stationarity wrt the correct target. Take for instance the Gibbs samplers or the Langevin version (MALA), which are usually not reversible and hence do not check detailed balance. They are nonetheless valid and satisfy global balance (1).
Thus the remaining challenge is to come up with a method to generate
these transition probabilities
Not really, since there exist families of generic MCMC algorithms such as random walk Metropolis-Hastings algorithms or Hamiltonian Monte Carlo. The challenge is more into calibrating a given algorithm or choosing between algorithms.
