I don't think titles directly transfer between companies, and yet the industry allows it. It's a very useful tool for advancement.

This may be true on some corners of the industry, but at the more competitive end (both in terms of competitive pay, and a competitive pool of candidates)... I believe it's common to relevel on hire. I've seen folks go from director to senior and from senior to junior at my org. The candidates being offered those seemingly big "demotions" often seem to be somewhere between unphased and enthusiastic about the change, presumably because the compensation package we offer at the lower level beats what they were getting with an inflated title and because they know their inflated title is nonsense and they're frustrated with the other aspects of organizational dysfunction that accompany title inflation at their current company.

What you say is real, and sometimes a promotion in one org can help bridge you into an org that would have been hard to get hired into as a junior, or harder to get promoted in. It's not without risk though. All things being equal, I'd much rather spend my time working on a strong team and learning a lot and being challenged than to be in a weaker org that's handing out inflated titles. Getting gud isn't a guarantee of advancement, but it's at least as reliable over the long haul as title inflation.