Like many things, it depends. And it depends on what kinds of things an engineer/scientist is studying. Here's a short list of examples from aerospace/propulsion applications:
Diesel injectors, liquid rocket motors -- the pressure is so high that many of the flows are trans-critical or super-critical. This means they deviate pretty significantly from ideal gas, and so we use so-called "real-gas" equations of state. These are usually based on cubic functions to approximate the properties. These work well away from the critical point, but at the critical point they all blow up. But, most of the time, there's very few places that are at the critical point and so we can get away with just ignoring it. To get critical points right requires... something. Molecular dynamics hasn't been able to do it reliably, and experiments are incredibly hard.
Air-breathing propulsion devices (jet engines, afterburners, etc.) -- the pressure is low enough that the gas is considered ideal. However, combustion and pressure changes through the compressor and/or turbine stages means the gas properties are variable. So, we use a thermally-perfect equation of state typically.
Low-speed aerodynamics -- by low speed, I mean subsonic and in the realm where cars to things like commercial jets operate. The working gas is usually unchanging (air, with no chemical reactions) and the pressure and temperature changes are small. So ideal gas is great. In fact, we can get away with calorically perfect gas and just hold all of our properties fixed.
High-speed aerodynamics -- this is for things that travel around the speed of sound to perhaps a few times the speed of sound. Here, compression starts to really matter and the gas properties are again changing a lot depending on the local flow regime. So just like in air-breathing engines, we use thermally perfect idea gas relations.
Hypersonic aerodynamics -- interestingly, once you go fast enough, the gas starts to take on constant properties once again. This is something like 5-10 times the speed of sound. So somewhat paradoxically, we can actually go back to calorically perfect equation of state and hold our properties constant. Of course, the constants we choose will be different from the low-speed case, but constant anyway.
Rarified gas dynamics -- when you have something like a re-entry body, strange things start to happen. The molecules of the gas might have different energies in different modes, and it is no longer an equilibrium. In these non-equilibrium flows, we have to resort to much more expensive calculations of energy populations. We can model these using multiple temperatures and can assign state equations for different energy modes. This is something you will not encounter in classical thermodynamics, but comes up in statistical mechanics/gas dynamics.
So that's a laundry list of things that I work on regularly and how we handle the EOS. You can see that for many things, the ideal gas equation of state works perfectly (pun intended) well. We can get fantastic scientific information using it -- not just engineering approximations. And often, when we want an engineering approximation to things, there are so many fudge factors due to uncertainties and safety margins that the errors from the "wrong" EOS are marginal.
However, it really depends on what you want to calculate. If you're doing studies are very high pressures, you need to account for the non-ideal effects. If you're doing things in rarefied gases, you need to account for non-equilibrium effects.
As a result, when I want to calculate heat, sometimes it's as simple as $\Delta H = C_p \Delta T$. Sometimes I need to solve the integral equation because the gas is thermally perfect and so the change in properties really is important. And other times, the EOS is non-analytical and I have to resort to numerical integration of lookup tables because it involves phase changes and exotic states.
Like all engineering work, we have to look at the totality of our assumptions and how much cost we are willing to put into a solution. If I need to design something to extract 10 MW of power, I can start with a simple hand-calculation assuming everything is perfect and if the math shows I can only get 1 MW of power out of it, odds are good it's not worth spending more time/money/energy doing further calculations. If it shows I get 11 MW, then it's worth moving to the next level of fidelity, for whatever that might entail.
