This is a tough problem to solve, but I’ll do my best to give you some pointers.
Bullet Journal’s main appeal is the physical nature of it that you alluded to; there are no distractions when you’re managing your tasks. It’s a separate, disconnected object with a singular focus.
That’s also its downfall, and as a digital person I ended up having to move away from it because I was frustrated by the annoying trade-offs. I wanted things to be searchable, I wanted to move things around easily, I wanted to be able to drill down into a specific subset of tasks, I wanted to be able to input new tasks without the friction of finding the book, finding a pen, etc.
If you’re willing to put up with how slow it is, and how fundamentally inflexible it is to have things locked up in ink and paper, or if you’re just charmed by the tactile impression and “old fashioned” lo-fi feel of it, then bullet journal remains an excellent system.
For me, it ended up being more and more frustrating over time, so I sought other solutions.
The winner has been Todoist. It does a few things that made it work better for me: it keeps track of completed tasks, and has a reward system built in (karma) that helps motivate you to not just complete tasks but manage them too. It’s also very clean, and structurally simple (unlike OmniFocus) so it can be anything from a basic list to an organized set of projects, depending on what you prefer.
The other main appeal is natural language processing. Adding a task is as easy as typing “answer David’s Medium response tomorrow #blog” (which is what I typed yesterday). Todoist parses it as a task due today in my Blog category. No menus, no dropdowns, nothing. Just typing what I need to get done and when.
And because it’s digital it’s immediately in sync across all my devices, so I’m never fumbling for a way to refer to my list or add something to it because it’s always with me in one form or another.
Bullet journal’s main appeal for me was the focus on accomplishments rather than stuff I needed to get done, and Todoist allows me to keep that benefit (in the form of Karma and a permanent record of what I did and when) without sacrificing the conveniences of digital and the more flexible structure of a traditional task management environment.
Task management is a very personal thing though, so I encourage you to try bullet journal and see if it makes sense for you. I admit that I don’t tend to struggle too much with notifications and distractions, so for me the fact that it’s on my phone/iPad/etc. doesn’t stop me from getting things done.
Good luck and remember not to overthink it!