Of course some talented and dedicated individuals will be able to pull this off, but it's not really a hobbyist-level activity yet. A couple key breakthroughs could help a lot, such as transforming plant somatic tissue with de novo shoot organogenesis. You could avoid all the tissue culture that is really the difficult part and start to transform less amenable species. Right now you're very limited to particular species and invested in long-term care of sterile tissue culture. Sterility will always be difficult outside of a lab.

Also, what's possible in this sort of setting right now is trivial; looking at selectable or color markers. Anything much more clever than that, and you're looking at tricky cloning, PCR, and sequencing which starts to get expensive fast. If you're making your own constructs, then you need the ability to extract RNA (not terribly hard, but phenol/cholorform is really helpful), make cDNA (kits are expensive, and hacked stuff is going to be a ton of troubleshooting), and pop it into a Gateway system or similar. This means you'll have to make your own competent cells, get some topoisomerase or a set of restriction enzymes and ligases, etc. etc. Buying the decent kits adds up to several thousand dollars, and an amateur might burn through that just troubleshooting their stuff.

Perhaps there is an analogy between biology today and very early stages of the home computer revolution, but I don't think it's a given. If sequencing and, perhaps more importantly, synthesis becomes accessible to anyone, then we'll see real progress.

I don't know much about the dialogue between amateurs and professional biologists, but I imagine a lot of the 'elitism' is just practical concern. Science is very very difficult, there's simply no way around that. There are just loads of things that we 'should' be able to do based on a read of the literature, but when you talk to people you realize that other's have tried and failed. Biology is complex, and sometimes success is a matter of attrition.