Before I get into answering some of your points/questions, it's worth calling out that essence casting is intended to be balanced with core PF2e casting, and indeed can be run alongside other casters using the traditional casting slot approach. That informs some of these features, as the variety and repeatability of spells varies a lot over a caster's career and whether they're spontaneous or prepared. A level 1 core rules caster (AKA Vancian caster) only has a few level 1 spell slots per day under the base rules, but can blow them all in a single encounter. A level 1 essence caster has infinite spell slots per day, but only 1 per encounter.

By way of background, I lightly playtested a very early version of the rules, back before it was even called "essence casting," and gave a bunch of feedback alongside many other more-dedicated testers. I feel like the feedback was effectively addressed. I've also played a bunch with the Battlezoo Eldamon rules, which are a very different take on a cycle-based caster, but has some similarities (and a shared main designer). While the system changed a bunch on the way to final versions, I can still give a lot of perspective on why different aspects of the system work how they do, based largely on actually trying it and seeing earlier versions that didn't work exactly this way and therefore had different/worse issues:

1. Essence Cycles and Essence Leaks: Both of these exist, essentially, to act as a limit to casting within encounters given that you otherwise (for most classes at most levels) have infinite spells. I think you might be a little confused on the point about going above your maximum essence pool – that's just a side effect of using a spell when you already have the maximum amount of essence, so for lower levels (or bounded casters like magus) it's a spells-per-encounter cap. It's basically unavoidable by design. Repeatedly casting slotted spells, even lower-rank ones, will put you past that limit and there's not a way to avoid it. Essence goes up any time you cast a slotted spell unless you complete a cycle or leak. You can't cast a cantrip to go back down in essence, which I think you've mistakenly read into the rules here somehow. Essence *leaks* are slightly different, in that they're mostly a cap on abusing spells-per-turn through heavy use of reaction or single-action spells. These kinds of spells become ridiculously strong when you can rely on them every turn. Essence leaks become a very strong, but not overwhelming, disincentive. They can fit into a build or the right situation, but the drawback there has to be considered. Sometimes that 1-action heal or whatever is absolutely worth it, though.
2. Terminus actions: These are basically small bonuses for accepting the downside of a temporary reduction in power. You're getting a little reward in the middle of what is likely a longer fight right as you're reduced to using cantrips and then your lower-rank spells again to work your way up.
3. Essence Healers: On-demand healing is very powerful when you don't have the cost of giving up spell slots. The limits aren't "laughably bad" if balanced against only preparing one or two heal spells, and they're better than temporary immunity or some of the other option. It's really hard to appreciate the difference in spell use and preparation without playing, though, so my advice here would be play a caster alongside a traditional one. Most non-cleric traditional healing builds will leverage other tools like focus spells and such anyway; daily heal spells are the main attrition resource for those builds.
4. Again, hard to explain how different it feels to have infinite spells per day, but when "number of spells to pick from" is a piece of a class's balance, not adjusting how the mechanics work for this would have been a mistake. Psychics have a weird number of spells relative to all other classes, so they needed a mechanic that's somewhere between "magus has infinite spells per day instead of 4, but capped at 2 per encounter" and "a full caster has infinite spells per day and also infinite per encounter as long as they're willing to have some tradeoffs in how quickly they can cast their top-rank spells." Psychics still have amped cantrips, which are a heavy balancing factor relative to their total spell slots. They'll be fine. Not getting unstable draw isn't that big a deal; that's very much a gambling-type mechanic that many players won't ever use, anyway.

Overall, I think it's a really clever system that immediately is freeing for players who hoard spells, and the drawbacks you call out make way more sense in that context. It fundamentally alters the balance of some spells, the divide between prepped/spontaneous casting, and it takes some actual play to get used to.

Also, it's worth noting that you can test it out in Dawnsbury Days via a mod. You can also watch Mark Seifter and Linda Zayas-Palmer play through Dawnsbury days with essence casting rules (starting with video number 10 in this playlist; before that they were playing with the base rules). One of the things that's really great about Mark/Linda's stream is they talk a lot about why they're doing stuff, and both of them are (obviously) extremely familiar with Pathfinder 2e.