No partner was sent the new Flash Next engine afaik, so Palm would have had the old mobile port of the existing Flash Runtime, which was pretty slow on mobile.
There was a lot of support internally to open source Flash, but the reality is that Adobe used a lot of partner solutions that have patents to improve Flash for various needs-- video DRM is one example, but there's other more practical features that used patents too. The work to remove all the patents would have taken at least 6m to a year, and that's far too much investment for no ROI.
Creating a throwaway account to comment as I worked with the Flash team when things came to a close...
Who knows if Flash would still be around today, but there was a great deal of innovation happening on the Flash engine up until the point the Flash Runtime team was effectively dissolved, because of Apple's decision. Up until that point, Adobe worked very close with Apple to recreate the Flash Platform from ground up with an entirely new engine. The new engine was vastly superior, and passed all of Apple's requested criteria for performance and memory usage on iOS. The runtime overhead compared to native was astoundingly very close, and effectively negligible for game development. In fact it was so efficient that most people internally thought that the new Flash engine could outcompete Java for general purpose cross-platform runtime based on several internal benchmarks. Even the language was improved with ActionScript4 (including Generics support) and things like threading/parallelism, simd, etc support was a focus.
However, everything went south when Apple when, despite the new Flash engine exceeding their requested requirements, Apple decided at the last minute that they are pulling out of the agreement. While it was not explicitly said, it was implied that Apple thought Flash support may hinder native game development that then could be sold on the iOS store. Once Apple pulled out, it was the straw that broke the camel's back as the massive investment into the new rewrite of the platform couldn't be justified. It was really sad as we had brilliant engineers and researchers working for years on the rewrite. But, sometimes that's life.