Eli5: Why does holding something even moderately heavy above your head for as little as 10-15 seconds cause so much discomfort in the shoulders?
Because the arms are supported solely by the shoulders which are not commonly used for static strength tests. The relatively small shoulder muscle group is designed and mostly used for supporting lifting objects up and not for holding things aloft. If you exercised those muscles harder and more frequently they would handle the load better and hurt less. But alas, you are weak and cannot shoulder the load.
If you feel as if you should have more than enough strength, it probably means that you don’t have appropriate mobility in your shoulder joints.
If your shoulder muscles are tight it takes more muscles to get them overhead in the first place, fighting against the tight muscles. Then, if your arms can’t get directly overhead it becomes more and more like you are holding an object out at arms length, away from your body.