Generally, learning through play implies that a child is using toys or props, interacting with peers, and/or moving around outside, as opposed to playing video games. While there are definitely benefits to learning by playing video games (hand-eye coordination, problem-solving, highly motivating), the use of computers is considered more of an engaging alternative to traditional instruction, rather than it's own kind of educational play.
Providing opportunities to play provides a host of benefits that video games can't touch, like developing fine and gross motor skills, encouraging kids to problem-solve and negotiate, build imagination and creativity, practice emotional regulation, and learning information in context, so that it's more relevant and therefore, more meaningful. The balance of work-play for kids varies as they grow, with youngest kids needing more play than older kids, even though it's helpful for kids of any age to have that opportunity. Employees of Google are encouraged to play, which allows room for innovation and makes for a more cheerful work environment. This guide, through UC Davis, has great information and tips on how to encourage play at every age level. However, how much play a kid should be getting is hotly debated, particularly as recess is getting cut more frequently in schools across the country.
Outdoor education is somewhat different, as it connotes structured lessons led by adults in the natural environment, activating all five senses as they interact with the world face-to-face, rather than via text. Outdoor education is somewhat new, and is considered a wonderful supplement to typical classroom activities, but it's not something teachers are trained to do or required to do. And you're right, there's a lot of evidence about its efficacy, but most schools are so pressed to deliver the minimum requirements, that they won't attempt any "extras." Although in Europe, spending time outside is fairly standard.
Regarding funding for technology, the idea is that all students at this point are going to have to know how to use various forms of technology in the future, and they should become fluent with navigating systems now. At the same time, technology is an engaging way to practice basic skills (especially the ones that you and I used to practice on flash cards...ugh), and playing games on ipads can actually teach them to get comfortable with the computers that they'll later use to send email, type essays, or design an object to print in 3 dimensions - and who knows what else? Probably more relevant is the fact that technology is a part of the common core, so schools have to be teaching it, and schools are receiving grants specifically for technology. There may be some grants available for outdoor education, but they are certainly few and far between.
I hope that answered your question. Thanks for the opportunity for a fascinating discussion!