I think you are burnt out and you desperately want this person to be able to get up to speed and take some of the pressure off you. Time off sounds well needed/earned. Please push back on your workloads - work burn out is awful.
I worked at a place that trained everyone (everyone!) in management, incl entry level juniors (equal opportunities etc) but their approach has stuck with me for 14/15 years for being so different (and albeit slightly uncomfortable). But, in this position, they would encourage a “what could YOU have done approach” - based on the fact we are often set in our ways, so you can’t change someone else but you can change your approach. You are measuring their success against what looks good to you, but what drives them?
For example, you say about recurring tasks - if they aren’t good at making lists, how about an excel how to with tickbox steps? Something that if you get more headcount you can resuse. Or, do they need a reminder? Eg on slack you can set reminders, or on their own calendar. Or do they prefer pen & paper? So get a calendar and pop down each Monday the tasks that need doing without fail. Yes work upfront for you, but it’ll save you time when you have to check their work.
As for notifying about AL, you could propose a calendar like others have suggested (on the wall or digital). And set in progress a process that a handover must be completed 2 days prior to AL. How far in advance are you booking your training? It’s sounds quite informal so putting all the sessions in ahead of time would be beneficial for both of you. And, maybe ask them how AL would work best for them eg calendar reminders etc and find a happy medium.
I appreciate it’s tough and feels like more work, but honestly, set them up for success will reduce your workload (and likely stress) - and also show management skills to superiors. Approaching it with a how can we solve this rather than heres 25 issues will be good for both yourself and your colleagues - and they might just recommend sometime saving tips for processes you’ve had in place forever because you think it’s the best/fastest way.
And I have to remind myself of this often, but remember, you never know what someone is going through outside of work/in life, and sometimes there is nothing at all they just want time off - and that’s fine too. Maybe us workaholics could learn a thing or two from them!