Can bad capacitors on a motherboard overload and cause boot sequence issues/random shutdowns?

As gregg wrote, the direct answer to what you asked is "yes". However, it's "yes, but", with a big but (just so your computer isn't offended, I'm not claiming your computer has a big butt). Let me provide some perspective and context.

The type of capacitor that's prone to failure is the little electrolytic cans. They do fail, and when they do it's visible, usually on top because the tops are scored to allow them to fail "safely". A capacitor can be in the process of failing, but not have visible signs, yet. They do age and change internally until they reach a point where they die and are physically obvious.

A capacitor is part of a circuit designed to operate in a certain way; the designer wouldn't spend the money to put them there if they didn't serve an important purpose. Where electrolytic capacitors are used, the circuit typically has a lot of tolerance because the capacitors do age and their characteristics change a bit over time. But at some point, they will be out of tolerance and then dead.

Good quality capacitors should last for many, many years. But there have been problems in the industry periodically where batches of crappy capacitors got used, and failed early in the computers' lives. So computer age isn't a reliable basis for considering or excluding capacitors as a potential cause. But if the computer is very old, the chances of capacitors being an issue is higher.

When capacitors go out of tolerance or fail, it changes how the circuit works. Depending on what specific function the capacitor is part of, that function is likely to not work as designed. So a wide range of symptoms are possible from failing or failed capacitors, including boot failures and shutdowns.

But boot failures and shutdowns can be caused by endless different problems. You've already looked at a number of the possibilities. Blown capacitors are an easy thing to check because they are readily visible. Ones that have just aged out of tolerance are a different story. That's the kind of thing that's the province of refurbishers; they have test equipment and detailed circuit specs to track down board-level failures in a practical way. Very few computer repair shops attempt board-level diagnostics or repairs because they just aren't equipped to do it. They swap boards with a new or refurbished one and send yours off to a refurbisher. So if you don't see visible failures, there isn't much you can do to determine whether capacitors are the problem.

So the answer to your question is yes, it could potentially be bad capacitors on the motherboard. But if they aren't visibly blown, that factoid won't be very helpful. Unfortunately, so many things could be the culprit that a Q&A site can't be a lot of help in solving the problem.