I love this idea! I can't tell you how many friendships I've lost in the past (or ones that could've taken off but prematurely died) from getting stuck in the "I think they don't want to talk to me"/"They think I don't want to talk to them" cycle. Then you end up with neither person initiating contact, and the whole thing just kind of ends. Many if not most people these days don't want to initiate, including me, so if you wait on them to reach out to you, they simply never will, even if they want to talk to you, or would enjoy a conversation.
I've been making steps toward being the person who initiates all contact, and so far it seems to work better than my previous strategies of "never initiating contact", and "initiating 50% of contacts and assuming they're not interested if they don't initiate". The only problem is I'm really not a natural at reaching out, so if left to my own devices I wouldn't do it often enough to maintain the relationship. I like OP's idea of externalizing it with a tool, that makes it a lot easier to maintain the kind of consistency needed to grow a relationship.
To all those saying this is forced or inhuman or robotic or whatever... Listen it would be great to have some kind of easy natural friendship where both people would do half the work, think of each other regularly and reach out in an easy non forced manner, but in my life experience that kind of relationship seems to be very hard to find, if not impossible.
In the real world it seems to me like the choices are be lonely and alone, or to do all the work (especially in initiating). In the past I chose to be lonely and alone, maybe holding out for that kind of unicorn, ideal friendship that people in this thread are talking about. I don't think it was a good choice. I switched, I'm doing the other approach and now I have people I can talk to. I do feel kind of resentful for having to put in way more effort initiating than the other person in these relationships, but it beats being alone and lonely, so I'd still say it's a net positive overall.