I kinda assume phones are going to be battery powered for the foreseeable future. "Gaming" phones with better cooling do exist, but they are a tiny niche. Most local AI users will want to serve their inference needs through a very different kind of system.
Yes, but the battery tech itself is improving. We're already seeing new phones approach 8000 mAH internal batteries, which is large enough that you can splurge on compute and still have some left over at the end of the day.