We know that $T_c$ is the temperature above which no amount of pressure could force a gas to liquefy.
But why is this? Somehow I don't buy the point that the gas molecules exert too much pressure back to get close and turn into a liquid. If we had tens of thousands of atmosphere pressure (such as on the inside of hot planets), we should be able to liquefy any gas at any temperature.
Your description of critical temperature isn't quite right.
If you increase the temperature of a liquid beyond the critical point, the atoms are moving so quickly that persistent structure fails to form and so you have something that behaves a lot like a very dense gas.
Similarly, if you increase the pressure of a gas beyond the critical point, it becomes very dense so that it's like a liquid but without persistent structure.
In other words, it's not so much that the liquid phase does not form, but rather the liquid and gas phases become indistinguishable (rather intuitively) and you end up with what's called a supercritical fluid.
Here's the phase diagram of CO$_2$ for clarity:
As you increase the temperature of a liquid, it expands. Its density decreases.
Similarly, if you take a gas, and you increase the pressure, it compresses. Its density increases.
Imagine that you take a container, that contains both liquid and gas. They are both at the same temperature. The pressure of the liquid is slightly higher, because of the force of gravity.
They both co-exist because if some of the gas became liquid, the pressure would decrease; which would cause some of the liquid to evaporate. Similarly if some of the liquid became gas, the pressure would increase, causing some of the gas to condense.
So, they are both co-existing, but they have different densities, and viscosities.
Now, take that container and gradually apply heat. As the liquid heats up, it expands, taking up more space, becoming less dense, and less viscose. As the gas heats up, well, it cant expand, since the liquid is taking up more and more space. So, the density of the gas is increasing, and its viscosity too.
At some point, the density of the gas and the liquid become equal, as are their viscosities. In fact, everything about the gas and the liquid becomes entirely indistinguishable. The boundary between them disappears. The container at this point contains only SCF ("supercritical fluid").
