Yeah, this isn't a thing that done by old engineers. It's a thing that's done by diligent engineers. I'm in my late twenties. Also, people have used computers to simplify work since they became available. I have my grandfather's punchcards that he used decades ago for p-delta checks on bridge structures sitting somewhere.
Confirming things by hand, or doing preliminary sizing by hand (where 'by hand' could include spreadsheets or things like mathCAD) doesn't take a lot of time.
It's not a question of not trusting computer programs. It's a question of not fully trusting anything. Fear is an engineer's friend. It shouldn't keep you from taking action, but it should be there in the back of your head reminding you of the consequences of what you're doing. You have to understand the action of your structure and be able to take responsibility for the actual math that went into it. With experience you can get the understanding of the structure from a computer model. That's great, it speeds things up and can let you do things that aren't reasonable by hand. There are two places where there are issues though. In the first case, you could have a situation where you become complacent and treat it as a black box without actually understanding the structure. This is dangerous. In the second case, there could be an error in the way the actual math, model or assumptions are implemented.
You have to be able to look at the structure and the results and be completely sure you understand why the reactions and stresses are what they are in each component. If you ever hit a situation where you look at something like the force distribution in a structure and don't immediately understand why it looks like it does and your response isn't to sit down and figure out why it looks like that, you're not actually engineering anything. If someone asks you why some critical load or reaction is what it is, and your answer is that the computer reported it, you've done something wrong. That's the first situation above.
In the second case, it's a question of verifying because it's not a perfect world and people screw up. It can be anything, but you have to be more careful with things like computer software because you didn't actually write it. If you don't check it, you're implicitly taking responsibility for the work of the software company. If your structure breaks because of an error in the software, they're still going to blame the engineer.
It's not like anyone goes and does a computer model and then checks it with a check of all the load cases and using all the special case code formulas on a piece of paper. You're going to use tables and simplified assumptions and formulas that you've simplified from the code or from other sources. I, personally, like to put pen to paper as much as I can, but it's personal preference more than anything. Nobody's going through the world checking things by doing hand checks for the code check on a beam-column using the full steel interaction formula on a piece of paper for every item in the structure. First off, people are really only doing math for items that could reasonably be critical. Secondly, they're either using a spreadsheet that they've verified or they're using a simplified formula (My/Myr+Mx/Mxr+C/Cr at the simple end, or incorporating as many additional factors as are reasonable) and then seeing if they're close enough to call it okay. If a case exists where it's reasonable to use the full code formula and you're doing things on paper, you're just doing it in the case that actually requires all the factors.
Doing enough math to make sure you're not getting a catastrophic failure and that your numbers are reasonable, when combined with some degree of judgement takes very little time. Hell, proper checking generally saves time because you're going to catch things that would be difficult to fix in the field.
Also, nobody's including any of this in their calculation package. Generally, a calculation package is the minimum necessary information required to satisfy people that the code has been appropriately applied and adhered to and that some degree of due diligence has been done. Additional information is confusing and takes time to prepare. I'm not going to write paragraphs in my checking calcs or my preliminary calcs to ensure it's clear to a code checker what my assumptions are. I'll write in simplified explanations that would make sense to another engineer that took a proper look at them, but anything more than that's a waste of time.
This is ignoring the things I'll do entirely by hand because it's just faster that way.