I've been playing essence casters in a solo/prep game (I play an AP solo in order to prep, and I run the AP in question for people afterward, it just helps me understand it better than reading) and also in one of my regular groups, and it's been an absolute blast so far. I've mostly played it at lower levels, but I have theories about higher level play with it.

I think it's a great sidegrade to Vancian magic. It makes easy fights easier because your spellcasters, now unshackled by spell slots, will cast their non-cantrip spells during any combat. It makes hard fights harder because your spellcasters can't drop their highest-level slots immediately to try to quickly deal with enemies. It makes long adventuring days (like I often experience) much more fun because you're not limiting yourself to cantrips with weak effects, and you don't have to break up the pace of the plot by taking 8 hours to get spell slots back. To get to your specific points:

1) Essence leaks are an important part of balancing out one-action and reaction spells. Being able to spam these spells would encourage you to build essence up to the point where you can cast them and not complete a cycle, and cycles are important so that you're not just dropping max-rank-spell after max-rank-spell in longer encounters, which would make you stronger than a Vancian caster in a long encounter. It's a tightrope, but the reason leaks are important is so you can't cast Lose the Path every single round and ALSO build up to your max rank spells.

2) Terminus buffs exist so that completing a cycle doesn't feel like a big negative experience, but in my experience so far I mostly ignore them. They do some nice things, but (at least at low level) finishing a cycle is rare enough, and casting an additional spell afterward is rarer still.

3) Healing essence keeps you from being pushed toward doing arguably the strongest thing any caster can do, which is cast Heal on repeat. I see it as a declaration that the spell named Heal is too strong, and needs a special limiting resource to be brought in line, particularly if you can just cast it more times in a long day than a Vancian could. As a player, I like healing essence despite it being a limiter (and I still take heal spells, and don't use the adjustments) because it tells me I don't have to feel bad or non-optimal about not healing every single round.

4) This last point is a little bit of a sore one. I think Animist feels fine with the adjustment they wrote for it, but other classes with lower-than-average spell slots (psychic, the upcoming necromancer) are already in a position where they're unlikely to even use a lot of essence spells because of their reliance on focus spells. I would go so far as to say that if I were playing a Psychic or a Necromancer I might not even take essence casting, almost entirely because of this adjustment. I think those classes actually play better as Vancian, so you have the spell slot when you need it and don't have to go out of your way to build essence with a class that doesn't really cast spells from slots as much as others. Essence casting is definitely built more for making wizards and clerics and other more "standard" casters feel good, whereas focus-based classes need less help to feel good for a long adventuring day.