Stay or leave for PhD
Since you say you like the research where you are and you are fairly productive there, you can probably successfully get a PhD there. It sounds like your advisor is not perfect (busy, etc) but few advisors are "perfect" from their students' perspectives. And if you stay, you won't have to interrupt your research to apply to other schools and transition to a new group. Staying where you are is a "safe" choice.

If you go somewhere else, some of your weaknesses (which you are able to ignore in your current position) may be exposed, and you'll have to work on improving them. And you will probably meet new collaborators (especially if you go to anther country) and be exposed to new ideas and techniques that will be difficult to learn, but will make you a better and more capable researcher.

So the answer to your question depends on what you want to gain from your PhD: Do you want to get a degree and some nice publications? Then staying where you are sounds like the easiest way. Or do you want to improve yourself and broaden yourself as a researcher? Then you might be better off leaving.

Having said that - of course, even if you stay, you can still challenge yourself and improve yourself and your capabilities. But since you don't have to "prove yourself" in your current position, there's nothing forcing you to do so - you'd have to be exceptionally self-motivated and disciplined.