If the eccentric part of motion is what gets you sore and, in turn, drives your gains (at least, hypertrophy-wise), should I care at all about the concentric part? What do I need it for anyway? For example, if I do lying tricep extensions, should I really pay attention to how I get those dumbbells up as long as I am mindful of how they go down (slowly)? Come to think of it, maybe it's actually beneficial to do as many raises as possible (and you do it more times if you "cheat") because you'll get more eccentric descents which means more soreness and more muscle growth (I'm not so sure about strength gains, though). Is my line of thinking correct?
If the eccentric part of motion is what gets you sore and, in turn, drives your gains (at least, hypertrophy-wise)
This is not a correct assumption. Muscle soreness is not a valid predictor of hypertrophy, and so while the eccentric part of exercise is mostly responsible for DOMS, that doesn't mean that it is mostly responsible for hypertrophy. In practice, both eccentric-only and concentric-only exercise cause similar amounts of hypertrophy.
should I really pay attention to how I get those dumbbells up as long as I am mindful of how they go down (slowly)?
You don't need to lower the weight slowly in order to work the muscles eccentrically. You just need to control it using the target muscles, rather than stopping the weight at the bottom by slamming it into the ground or your body. In fact, lifting faster and using momentum is arguably better.
David Scarlett is correct in that soreness is not an indicator of hypertrophy but I do think that it's worth calling out that there are knowledgeable people who utilise and recommend "cheat" movements in particular circumstances.
For example, Jeff Cavaliere of Athlean-X (a YouTube Channel) recommends the "barbell cheat curl" in a video that is summarised in this [Men's Health article] (https://www.menshealth.com/fitness/a30687831/athlean-x-biceps-arm-workout/).
The summary in this particular case is:
"We want to take advantage of the fact that our muscles are stronger eccentrically than they are concentrically .. So if we can get that weight up via a little bit of a cheat, then we can overload the eccentric, which is a known stimulus for hypertrophy"
Use the cheat curl sparingly in your training, maybe twice a month.. "You don't want to abuse this move.. Used correctly, it can overload your biceps. Used too frequently, it can create shoulder issues because it does involve your shoulders more than you think when you're cheating up the lift."
From personal experience, I like the barbell curl exercise but will sometimes cheat the last rep or two if I am getting fatigued and want to add a little extra volume. In these cases, I tend to perform the eccentric even more slowly - but this is very possibly more psychology than science, since part of the drive to do so is to convince myself that I've not missed out too much by cheating the concentric a little!
More details in another Men's Health article here. And I'm sure that I've heard AlphaDestiny (another YouTube channel) talk about using a little cheating on dumbbell rows and that is echoed in this T-Nation article (under "Good cheat exercises").
To conclude: cheat reps have their place but you need to think carefully about how and when to incorporate them and you need to make sure that you're doing them safely since they are, by definition, more weight than you can handle.