"It would burden them"
There's a version of the belief that isn't about their capacity but about the telling itself. Whatever their plate, the worry is heavy, and handing a heavy thing to someone loads them with it. That's what burden means.
Michael Kardas, Amit Kumar and Nicholas Epley, in 2022, studied what happens when people have deeper conversations than they'd normally risk. Across a series of experiments, people expected deep conversations with strangers to be awkward and expected the other person to care little about their disclosures. The conversations turned out less awkward and more connecting than expected, and the reason was that the other person cared more about what was disclosed than the discloser had predicted. The gap was largest for deep conversations, and it narrowed when people talked to close friends and family, whose care they already knew about. That last finding is a limit, and I'll say so. The study is mostly about strangers. But the direction is the one to hold. People expect their disclosures to be received as an imposition. They're received as interest.
Consider what being told actually is, from the listener's side. Your sister learns that you've been waiting a fortnight for a scan result. What she receives isn't the scan. She can't have the scan. What she receives is that you chose her, out of everyone, to say it to. That you trusted her with the private version. That's not a weight. It's about the most flattering thing one adult can do to another, and most people, on the receiving end, feel it as closeness and not as load.
There is a weight, and I don't want to pretend otherwise. She'll think about you on Thursday. She may sleep a little less well on Wednesday. But she'd have been thinking about you anyway, because sisters do, and the version she was thinking about was a version she couldn't help. The worry you'd be handing her is smaller than the worry she'd have carried on finding out afterwards that you'd been alone with it.
"It would burden them" imagines the telling as a transfer, like handing over a heavy bag. It's more like turning on a light in a room they were already standing in. They were in the dark with you. They just didn't know what the room was.