I think some of this is conditional on how happy/unhappy you are in your predoc. I'm a second year predoc at a T3. From what I've seen, outcomes for second year predocs applying are usually much better than first year predocs. A lot of this is because your letter writers can say more about your research ability with an extra year of working with you. You mention being unable to pursue your own research. Is this entirely because of your workload? If not, I would encourage you to try to create some sort of research discussion group with your cohort. We have this in my program (set up by another predoc) and it is incredibly helpful to just have an hour each week to discuss research with your peers.
Your predoc program might also be more rewarding next year if it is in person. I know a lot of first year predocs in my program haven't been able to take advantage of a lot of opportunities (seminars, job talks, connecting with other predocs/grad students, free food, gym access etc.) because of being fully remote.
I know that if I was a first year in my program with a T12 acceptance I would stay for a second year. But I also like my cohort, can take grad classes in my program, and have a supportive advisor who allows me time to pursue my own research if I need it. My predoc program is valuable to me outside of just being a positive signal for my application so another year is not really a "waste".
Another thing worth considering is that if you got shut out of most of the Top 10s coming from a Top 3 RA position (with presumably good grades), it could very well be the case that your letters were nothing special. One significant but rarely spoken of downside to RA-ing for a renowned faculty member is that these people usually have a larger frame of reference of what a good candidate is, and therefore, your RA work might be seen as adequate but not outstanding enough to warrant a compelling letter (which is usually the tipping point in Top 10 admits, apart from the significant randomness/luck). If this were the case, there is little to no merit to staying an extra year.
Furthermore, there's no guarantee that you'll place as well as you did this round; although I'm pretty certain you'd be able to land a comparable or slightly lower ranked admit next year. Therefore, if you are fine with a Top 15 or Top 20 admit (worst-case scenario) next year for another shot at a Top 10, it might be worth it to try again next cycle. Personally, I don't think it's worth it.
You can also speak to the faculty member you're RA-ing for, and get their thoughts on whether or not you should put off grad school for another year to try for a higher ranked school. If they are ambivalent or against it, then it's a signal that their letter wasn't very enthusiastic to begin with.
I'll elaborate more on what I said because I think I may have been a bit too harsh, though what I will say at first will actually sound harsher: I think it's possible that you feel entitled to being accepted at a top 5 program and that you have yet to adjust to the fact that this not feasible this year. I do not think that your reasoning is rational. First, you have no way of being confident you'd be admitted to a target school next year (that's hard in any year, could be your LOR just isn't that great). Second, as others have pointed out, placements from top 5-20 programs are basically comparable to those from top 5 schools in terms of where top students place (the network benefits are there, peer effects less so, pedigree less so - latter two are non-trivial but do not seem like dealbreakers to me?). Your education won't be different, your stipend will be comparable etc. So not sure it makes sense to take an extra year on the basis that you could get into MIT or Harvard next year.
If you want to take an extra year before embarking on a PhD for personal reasons, that makes a lot of sense to me. If you had a problem being accepted anywhere or were woefully short of your target, delaying a year makes sense to me. If the department that admitted you is a bad match, it also makes sense to wait another year. There are lots of great reasons to apply next year. I do not see any of these reasons reflected in your post. I see you exaggerating research findings in a way that is basically absurd, saying you are tired of being a RA, don't want to apply again but, whatever, some table buried in a paper is sufficient to justify it - what?
I do not want to crucify you here or anything. Being rejected sucks, not achieving your objective is miserable. There is something almost humiliating about failing to get into your target school. You spend 3-6 years of your life aiming for it and it doesn't happen? It starts making you think pretty irrational things. In your case, you got into a top 12 program during a really bad year but it clearly isn't what you wanted deep-down. However, you still applied there, presumably because you'd be content going there, because it has something to offer etc. So is it actually a poor outcome or is it just suboptimal relative to your ideal scenario?
If I have advice, it's to think this over to think about what you actually want here, meditate on it etc.
I do not mean to say something personal about you, contrarily I believe you are honest and kind person, but this top 5 - top 10 thing is an obsession. Life is not in a perfect hierarchy as we sometimes think of it. People move up and down and there are lots of factors going on that you cannot optimize. If you are a good fit with your T12 as you have said, and you love the place/school, move on! Do not exhaust yourself with all the pain and suffering. You can be a good economist anywhere and people will respect you if what you produce is respectable; these apply even more if you are also in a good place, and you are!
I think you really overestimate the differences between a school "ranked" 7 vs 12. These are all great programs. They all have faculty that care a lot about graduate students. They also have faculty that don't pay that much attention to graduate students. You want to think about whether the best offers that you have are good matches for you. Are they strong in your fields of interest? Schools in this range aren't great in every subfield.
Why did I put "ranked" in quotes? You might be surprised to learn that the rankings are not based on placements, research output, etc. They are based on a survey sent to all department heads and directors of graduate study and ask responds to rate every program in the United States on a 1 to 5 scale. It's largely a popularity contest and is very slow to react to substantive changes in programs.
Is it better to be a star at a number 12 program or middle tier at a top program? It's entirely about the quality of your dissertation, not the name on the jersey, so to speak. Your outcome is almost entirely determined by your own work, effort, etc. The program matters because it feeds into your work, not because you on the placement list at Yale vs Penn vs Michigan. Pick the program where you think you'll do your best work.
Thanks all for the perspectives and comments, including the "harsh" ones. All great ideas, I will schedule a one-on-one meeting with my boss once I have all my results. I think, absent my boss saying something new, I will commit. Also I have some theory interests so maybe it is beneficial to begin grad school when my math skills are still relatively fresh from undergrad.
Just one more piece of information that may be of pertinence: I received ~85% percentile on the GRE quant (I took it once and didn't put too much focus as I was going through some personal issues).
(To give more context, I went to a T15 undergrad, with a ~3.8 GPA, took hard math classes (think point-set topology and PhD-level measure-theoretic based probability theory) but also got a bunch of B's (though I got A+ in Real Analysis I and point-set topology).)
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