I suppose I wondered if friendships that floundered because i didn’t bother were genuine friendships Or whether me making the effort was actually more of an annoyance to them?
Well, from my point of view, 'genuine' friendships are conducted purely according to what the people involved want from them, which may of course change over time. If a friend gets in touch with me, I don't see that as them 'putting in effort' or doing me some kind of favour, I see that as them wanting to see me or talk to me.
If that person felt they were putting in an undue amount of effort into the relationship compared to the pleasure they got from it, or was keeping some inner ledger of who got in touch with who most, I don't imagine the friendship would survive for long, because we would have different, clashing philosophies of friendship.
For instance, I have absolutely no idea who usually initiates contact in my friendships. If it is me, I clearly don't feel it's a burden, or that I'm 'putting in all the effort', or I'd be conscious of it as a negative. But I would in any case only be contacting the person because I wanted to hear from them/see them, rather than out of some sense of duty, so the only person I'm doing a favour to is myself, if you see what I mean.