Equilibrium and steady state are often misunderstood.
Equilibrium: properties are uniform throughout and they do not vary with time.
Steady State: properties at specific location do not vary with time.
Why only state of equilibrium is used to do thermodynamic analysis or modelling of system (at least undergrad engineering thermodynamics course do)? Why not use steady state instead?
The question you're asking is very much legitimate. In short, equilibrium is used as a condition, because that is what the well-established mathematical apparatus of equilibrium statistical mechanics (and thermodynamics) can handle. In contrast, it is much harder to deal with non-equilibrium steady states or the approach to equilibrium. Nevertheless, non-equilibrium statistical mechanics (and thermodynamics) does exist, and concerns itself with the approach to equilibrium or even non-equilibrium steady-states.
A bit more precisely, when a system is in a state of thermodynamic equilibrium, what is really meant is not spatial uniformity but that it is described by one of the standard statistical ensembles: microcanonical, canonical or grand canonical. These are what you would find in a textbook on statistical mechanics, and with their assumptions you are able to handle a large variety of physical systems: as long as they are in equilibrium.
As you probably recognise, in nature equilibrium is more the exception than the rule. Many physical systems may be approximately close enough to equilibrium that you can use standard tools, but many are in fact not. A simple example is a wire under voltage bias and current flowing in it: even on short time scales, when the heating may be negligible, and it looks like it is in a steady state, the charge carrier distribution in the wire is far from what the equilibrium distribution would be if you just let the isolated wire relax. A very complex example is life: living systems are actively using energy not to equilibrate (their temperature, particle number distribution, etc.) with their surroundings. Out-of-equilibrium phenomena (especially when quantum; everyone loves quantum) are in fact an area of intense current research and you'll find tons of new papers appearing every week dealing with them. I'll leave you with the Wikipedia page, it also looks like it has a decent amount of recommended resources. Hope this helps.
