Per this, your statement isn't fully correct either, since the new rulings made the USPS an outlier.
It was misleading of Bishop to bring up pensions in your linked politifact article[1]
Take the case where every company in the USA (including the post office) is required to do X and Y, but only the USPS is required to do Z. It's a true statement to say that "only the post office has to do Z". It's also a true statement to say "Only the post office has to do all 3 of X, Y, and Z". But the latter statement is abusing boolean logic to make it sound like X and Y only apply to the post office.
[1]: It's also misleading to say they "must fund 100% of the costs in advance" when the actual requirement is that the pension plan be "fully funded". If you're fully funding your pension plan, compound interest is probably covering at least 50% of the costs.
In addition to the reasoning you posted, this is factually incorrect. Every single private company in the US is legally required to prefund their pension plans (there's some complex laws handling the actuarial stuff and interest rates, eg MAP-21). It's so incredibly wrong to say that the post office is an outlier because they're required to. The post office was an outlier until they were required to prefund pensions. It also muddies the waters to conflate retiree health insurance with pension plans.