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You actually want https://www.measuringworth.com/uscompare/ for the index – as https://www.measuringworth.com/uscompare/relativevalue.php which is the calculator results page. (And it was coded by an Ohio State University college student when it was on EH.Net, before being maintained by me as a college student, before being transferred over to Sam Williamson's personal-business site about ten years ago where it is now – a transfer which I always thought was weird but that's academia for you apparently... regardless, it explains the code quality.)

Yes it is stable, inflation is a small steady largely predictable thing, that's what stability is, predictability. By contrast, Bitcoin is highly unstable, no one has any idea what it's value will be in a week, let alone 30 years down the road, but due to the dollars stability you'll have no problem finding 30 years loans denominated in USD. The USD is the very definition of a stable currency. Stable doesn't mean never losing value, it means being predictable enough to plan for the future.