Zone 2 is one that took me a loooong time to come to terms with in my run training. Like you say, it just doesn't give you that same level of intense feeling you're used to. It just serves a very different purpose. Its benefits come into play for endurance training mostly (there's other benefits, but this is the most tangible one). If you're wanting to train for a half marathon for example; If you go out hot every day you train and run in threshold zones, you'll probably never build the capacity to make it much past a couple miles before you gas out.
Those zone 2 workouts are meant to be long. If you stick to that zone, you can hit longer endurance targets and actually build capacity for going longer with more intensity as your training progresses. We could go into a whole other tangent about mitochondrial adaptation and fat metabolism/metabolic efficiency benefits of zone 2, but I need to not write a novel every time I respond here.
Similarly, zone 3 has most of its usefulness in the endurance world. While zone 2 is about building capacity without intensity (our ability to go further) with low fatigue, zone 3 is that next step up where we're starting to train/maintain intensity over longer periods. This is where I spend a lot of my maintenance runs. I'm not looking to build capacity when I go out for a 5mi familiar route, but I do like to maintain some level of ability to maintain intensity without the need for several days of recovery like going over threshold (zone 4 & 5) would do. It's a way to kind of toe the intensity line while maintaining high training volume.
Example "maintenance" run from two days ago:
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Compare to the "all out" effort from 1.5wks ago where I basically had to take the rest of the week off from running with only light gym sessions (this is more like race day only effort):
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So to compare to what you're talking about using this heart rate zones in short intervals: that's where they have less usefulness and benefit. High intensity interval training, which it sounds like is what you're usually doing, should be done going in and out of the threshold and anaerobic zones.