If you are checking the calculations by hand, then it seems to me you are verifying the software. It's not a bad thing to do, but I wonder why it's considered acceptable when the results don't match your own? If your results are correct, then you have a malfunctioning tool...

Wouldn't it be helpful to be able to look at the code to see what it is actually doing behind the scenes?

Not if you can't code, which most engineers cannot (nothing wrong with that). In fact, I'd say without a team of highly trained software engineers and a whole lot of time and money, having the code for a CAD program is almost useless. In civil engineering, it's my understanding that it's standard procedure to check everything CAD tools output. If a bug leads to failure in the field, at least one person, but probably multiple people, has failed horribly and he will rightfully be held responsible.

There are many problems where finding a solution is much harder than verifying a solution.

A well known one used all the time in crypto being finding the prime factorisation of a number versus verifying that prime factorisation.

So verify the solution you give me!

If the whole point if verification is that you don't trust your software, then that's not a very good idea. So you use the ultimate slow-but-reliable computer, the trained engineer.