I did my Engineering Physics I homework using mirrored writing. The instructor didn't mind. Looking back, it's a bit risky given you're drawing equations and arrows, not only text. I hadn't tried this before or since.
At the end of the semester, I told him physics seemed fun but I thought software engineering was a better career choice for me. Only years later did I find via LinkedIn that he had worked as a software engineer for ten years before. :D
"probably because it's more natural for a left-hander to write backwards".
No, the key is that pulling the writing instrument requires a lot less muscle than pushing it. This means that mirror writing requires 1) more muscle strain or 2) less muscle strain than writing with the dominant hand if either the hand is switched, or the direction (LTR or RTL). If you swap both factors at the same time, then the strain is the same.
Left-handers who write LTR and switch direction (RTL) reduce the muscle strain compared to their usual writing as they're now pulling the pen. A right hander who does the same in this situation has to push the pen RTL with their right hand, which is harder than they're used to. A right hander should use the left-hand for mirror writing so they pull the pen like they're used to.
From this information, you can derive why people think it's easier to mirror write by switching hands, and others think it's easier for left-handers than right-handers to mirror write with their original hand.
TL;DR: Yes, there is initial effort to start mirror writing. But the real difference is that in the long-term, one group will find mirror writing to cause more hand strain than normal, and others will find it less hand strain than normal, because in one group pushing the pen is normal and in the other pulling it is.