The problems were not with the email notification, which today worked just fine (except issue below), but with the need to access the forum by a browser to make new posts
Unless you want to write your own XenForo addon, it's extremely unlikely you will
ever be able to do that. What you could do in the past is irrelevant; that software either no longer exists (SparkList) or the feature would not be supported in the new version we were going to be forced to use in another couple of weeks (vBulletin). Unfortunately, I have to make decisions based on the real world, and not an ideal but nonexistent user utopia.
Stopping "watch forum" full-post notifications would result in my not reading the forum accept when I feel like browsing the web once a month... If the RSS feeds work properly, so that I get all posts, including posts starting new threads, in my mailbox (at least one Wordpress blog is delivered to my mailbox via RSS perfectly well).
You have exactly two, and only two, choices:
1) Continue using email notification in its current form. There is a (small) chance that at some point in the future it may be enhanced, possibly even including one of the features whose lack you say make it currently unusable / undesirable.
2) Do not use email notification.
3) There is no third choice.
Do you not keep copies of messages you send? Without copies of my own posts neither my own threads are complete in my local archive, nor those I replied to. Please do not substitute your own judgment of what customers should have for the customers' own.
I'm substituting my own knowledge of what is currently possible, for the customer's fantasy of their ideal world. What you want is not currently possible; there is no way to be emailed copies of your own posts. There is absolutely nothing I can do about that, regardless of whether I agree or disagree with your wants.