"They've got enough on their plate"
Start with the belief that does most of the work, because there's research that measures it and the measurement is one-sided.
Xuan Zhao and Nicholas Epley, in 2022, ran a series of studies with over two thousand American adults on what people expect when they ask for help. Across scenarios, remembered experiences and live interactions, people who needed help consistently underestimated how willing others would be to give it, underestimated how good the helpers would feel about being asked, and overestimated how inconvenienced the helpers would be. The reason, the researchers found, was that people asking for help thought about it as a question of compliance, whether the other person would grudgingly agree, and missed the other person's actual motive, which was that they wanted to help and felt good doing it.
An earlier study by Francis Flynn and Vanessa Lake, in 2008, found the same gap in the simpler act of asking strangers for small favours. People underestimated by as much as half how many would say yes. The askers were thinking about what saying yes would cost the other person. They weren't thinking about what saying no would cost, which is the thing the other person is mostly thinking about.
An earlier book in this series uses both studies for the habit of not asking for practical help, and I'm reusing them here because the burden belief is the same forecast pointed at a different request. When you decide not to tell your friend about the scan because she's got enough on, you're doing what Zhao and Epley's participants did. You're pricing the telling by its cost to her and forgetting her motive, which is that she'd want to know, and would feel closer to you for being told, and would feel worse, not better, to learn later that you'd carried it for a fortnight within reach of her.
"They've got enough on their plate" is a sentence about their capacity, made by you, without asking them. The research says that when people in this position are asked, they don't experience the plate as full. They experience being trusted. The belief isn't a kindness to them. It's a forecast about them, and forecasts of this kind run in one direction, and it's the wrong one.