Are you me? Haha I'm in the exact same boat. Although slightly older (35), I've always dreamt of owning one, been looking on and off for the last few years, DID give up at one point and bought something else, but am now back on the hunt and even more eager than ever. I also suffer from the same problem as you do, having something very particular in mind.
I know the feeling of trying to find something for $15k, too. I agree there are not a lot of great options in that range that fit the vision I have of my perfect FJ40. Some occasional gems for sure, but yeah few and far between. So as I've continued to search I've continued to save. This has kept me encouraged knowing that one way or another my budget and my perfect truck WILL intersect. I really am feeling closer than ever today.
Of course there is the option of buying one of those cheap beaters and transforming it into my perfect truck, which I would absolutely love to do if only I had the space for it. My current 1 car garage is a rental space that would not be suitable for a long term teardown and rebuild project. However in addition to searching daily for a truck, I'm also searching daily for options to change that. So I'm encouraged in this regard as well by realizing that I have the capacity to alter my life in order to better accommodate my goals, which in this particular case is to own and enjoy an awesome FJ40. As long as the enthusiasm is there, you can remain motivated knowing that you can and will do whatever it takes.
My gut tends to know immediately if a particular truck speaks to me or not, which I imagine is a feeling you are familiar with too. I agree those do tend to be quite rare which can be discouraging, especially when those gems suddenly turn into missed opportunities. What this has continued to do though is help me really hone in on what I know I want, what I know I don't want, and in what areas I can be a little more flexible. It's an iterative process that takes time, but I believe it will pay off in the long run so I am happy to remain patient with it.
One thing I will say that has definitely clicked for me in recent months which hadn't before, is a much stronger feeling of being absolutely ready. Previously I found myself being a little too wishy washy in my search, making more excuses than I needed to, and not really being "all in." Now that I have a confident amount of cash on standby, have learned more about what precisely it is I want, and have put myself into the mindset of being ready and willing to act on a moment's notice, those feelings of disappointment and frustration of the past have rallied into a newfound wave of excitement and optimism.
Interpret all of that how you want, but yeah I figured I would share my own experience thus far since it's something I think about every single day.
Obviously I'm still on the hunt myself so can't really offer any legitimate buying advice, but do want to mention a couple of key search techniques that have helped open up a lot more options to me now that my search has gotten serious:
-Craigslist search alerts - I have a search alert for "FJ40" setup for all Craigslist hubs within 1,000 miles from me, with an automatic e-mail being sent as soon as something comes up. There is some noise to sort through for sure, but it puts about 4 FJ40s in front of me every week.
-Autotempest.com - This is great for checking multi-regional listings on Craigslist too, but also more importantly, Facebook Marketplace. It remembers your search criteria, so it's very quick to go to the site and take a quick glance at a few neighboring cities every day or two. I have to be careful though because I also tend to see a lot of other awesome vintage stuff available on there which distracts me haha.
-This forum, obviously. I'm sure you are already subscribed to the classifieds page here, but yeah a lot of options are posted here regularly. I have another classic car which I bought from somebody on the enthusiast forum over there, and am hoping to luck out with the same sort of great experience here with an FJ40.
Again, just my two cents, but stay motivated and good luck!!