What practical steps do you take to recover rankings when Google hits your site but GSC shows no issues?
it depends on which update screwed you over. Each update comes with its own triggers, effects, and fixes, so you need to discover the triggers and effects in order to know the fixes...
Judging by the 2 screenshots it looks like both sites are neither sandboxed, or penalized. They're simply not good enough (SEO-wise) to rank anywhere. And the reasons for which I'm saying this is because:
1) the sandbox that my sites have experienced in the past doesn't look anything like in your screenshots. In my case, I've made the sites live, submitted the sitemap to GSC and for EXACTLY 9 full months (to the minute) neither site had seen any click or impression, no matter what I tried.
But your sites got clicks and impressions, so this makes me assume that it's not sandbox... at least not the sandbox effect that I've had, maybe there are new effects nowadays vs few years ago when my sites got sandboxed, so that's a possibility, but until proven wrong I will say that it's not sandbox.
2) and I'm also saying that they're not penalized, because I've just checked the dates in which Google have released major updates, and none of those dates corresponds to the date when your sites stopped performing.
Furthermore, both of your sites seem to linger around pages 5-6 of google, which tells me that google doesn't have anything against your sites (yet), otherwise you would have disappeared from top 100.
But here's a disclaimer again: when I'm saying these I'm doing it based on the experience that I have since few years ago when I was involved with SEO (nowadays I'm not involved anymore), because it's possible - once again - that things have changed in the meantime. I mean, OK! Things always change in SEO, that's for sure
But with regards to the graphs in your screenshots I don't think anything had changed, which is why I said that your sites are neither sandboxed or penalized...
If anyone thinks otherwise and can prove me wrong, by all means, do it! I will happily stand corrected...
Why do they start strong, then drop to near zero impressions?
I don't know, but this happens to many new sites. My guess is that google's testing your site's overall performance (UX, content quality, speed and technical issues, etc) against their traffic to see how the site performs, and based on this intel they either sandbox you, or let you to your own devices, or (if everything's good) they rank you in top 10 somewhere...
Is this a sandbox effect, or some kind of Google adjustment?
doesn't look like sandbox to me. But better wait for more opinions
What practical steps can I take to recover and grow them again?
simply work on your site as usual until it starts climbing up the SERPs. Keep publishing quality content and internally linking it with other relevant content, keep promoting your site outside of google, keep building links, all of that normal stuff...
Depending on which industry you're in, it might also help to become active on social media (Instagram and Twitter in particular) and to boost your EEAT scores (particularly important if you're a random nobody but you operate in a YMYL industry like health & medicine, finances, relationships, well being, etc)
