For my money, the issue isn't lack of cross-browser support. It's the process by which the PWA specs were designed in the first place.
There are more than a handful of cases across the past decade where a new "web standard" didn't emerge through the usual standardization process — proposal, discussion, experimental implementations, feedback, etc. Instead, these "standards" were developed directly in one of the browsers, shipped, and then promoted as a feature that developers should begin targeting, totally bypassing the usual process, with the tacit expectation that other browsers would be forced to follow. Apple did this with <canvas> and some aspects of CSS3 to help with their push away from Flash. Google did this with many the PWA APIs.
All along, as the PWA APIs have been released from the Chrome lab, there's been a certain contingent of developers who heap scorn upon the Safari team for not implementing these APIs too — as though the existence in Chrome (and, in some cases, Firefox, who tend to follow-fast as a survival instinct), plus the Chrome team dropping specs off at the W3C working groups, were cause enough for these APIs to be regarded as "standard". The error of these devs is confusing ubiquity and standard.
Most PWA features are, at current, about as standard as ActiveX was. They were designed by Google in private, to suit their own ecosystem goals. They might be good for the web, by providing an alternative to native apps — but they might also be bad for the web, by forcing browsers to serve as an app platform instead of a hypermedia user agent.
Sadly, we don't get to have that very important, very nuanced discussion. Instead, we're having the discussion Google wants us to have (quotes from this HN thread):
"Apple refuses to implement PWA features in Safari"
"There's a somewhat serious blocker for some apps in that iOS doesn't do push notification to PWAs"
"Apple purposely slowing down Safari development to make it an unappealing platoform for web devs"
"As I'm building a PWA, I'm finding certain Web APIs that were intended for PWAs to be incompatible on Firefox/Safari"
"the major problems are lack of Safari support"
Google won a dev-rel PR battle here. Apple have won similar battles with other "standards" (for instance, there's a similar split in opinion around Flash, but until recently the "Flash deserved to die" contingent have been dominant, which suits Apple just fine).
So why did PWAs fail to take off? I would argue that it's because they're a significant alteration of what the web is meant to be. They're a big step toward defining the web as a full-throated application runtime. I don't know that the rest of the web ecosystem is on board with that redefinition. At least not yet, and not merely because Google tried to will it into being. That change may come in time, perhaps under a different name, or perhaps not at all. But if it does come, I hope it does so through a process that reflects the interests of more than just Google.