I've had a chronic mercury poisoning, fortunately, after self-diagnosing (but I was unsure, I just had no other options so I pursued the only one that made sense) I found a university clinic researcher doctor who thought the data I had from lab tests supported starting chelation to see if it helped. I got DMPS injected. Initially he said I'm in a "gray area" and that there must be something else. He never found anything else, and the chelation provided miracles. For example, I nodule I had had in my double-normal-size right thyroid for decades, unchanging, within a few months completely disappeared. Fortunately I had just gone to an endocrinologist a short time before to reconfirm that nothing had changed, and after there was a lot of "activity" in that area for weeks, starting after chelation, I went there again. The endocrinologist put me under the ultrasound twice because he did not believe it, but the thyroid was normal size and the nodule - gone.
There is a loooooong list of other things, from RSI, dry eyes, warts on the feet, winter depressions, cold hands and feet, psoriasis, digestion, sleep issues (incl. nightmares - my dreams now are incredibly beautiful and interesting, wow!), occasional cramps, longer and longer colds that in the last years before I was diagnosed only stopped in spring and during winter just didn't want to go away, etc etc. which all completely disappeared!
Every doctor ever, and myself too until I was forced to seek an answer because nobody knew what was going on and just declared me "fine", thought that each of those many many small issues was either normal part of live, due to aging, due to office work, due to "head" (imagination/psychology).
Things like that I had significantly decreased mental abilities and was more easily stressed I only recognized when they disappeared with chelation. Note that I called myself "healthy" until the very last day of escalation, when I was forced to start looking into it, and just like OP got not help from numerous doctors who all just went "by the book" - which has nothing about chronic heavy metal poisoning so it doesn't exist. (Also note that I don't fault any of those doctors, I would go again to every single one of the many I visited if I needed to, but now knowing better how to use their definitely existing and very high expertise.)
But I wanted to answer specifically about your concern about chelators.
After DMPS injections I switched to DMSA orally when I felt small doses over a long time where a far better fit for the remaining (and ever-decreasing) issues. DMSA is a key chelator for lead, secondary for mercury.
First, do look at the profile of the chelator in question. Some EDTA variants indeed remove vital things, but chelators like DMSA actually only remove the "bad" metals - and zinc.
However: I took tons and tons of DMSA for years, 100 mg every 3 hours ("Cutler protocol" - but I did experiments to verify that that recommendation indeed worked best for me, I didn't just take anybody's word any more and did a lot of self-experimentation, and my doctor, all those in my country, only have experience with DMPS injections ("pulse dose")). And yet, I never had any issue with zinc. I substituted a little bit, but far less than recommended. So even on an extreme chelation schedule I never saw any issues for the one "good" metal that both my chelators removed.
If you only do "pulse dose" it is a non-issue to begin with. Only a tiny fraction of whatever metal the chelator binds to is available to get into contact with it, most is stored somewhere. They all only work in extracellular space, and even there only reach a fraction of what's there. The stored metal, both good and bad, don't just slosh around in the watery solution of your extracellular body water or at the walls, most of it is inaccessible to a chelator. So only if you don something as extrem as I did with DMSA is it an issue at all.
Pulse dose DMPS injections, when there actually is a heavy metal to be chelated, indeed are a little hard on ones body, for a day or two. People in an already pretty on-the-edge condition may want to avoid it. I found combining DMPS injection followed by DMSA "a la Cutler" for a week helped. Just Cutler was/is far milder, and you can adjust your dose downwards (or start very low to begin with).
DMSA in food grade is pretty cheap when bought in bulk from China, who are the main makers of that chemical (Russians invented DMPS in the 1960s, but other countries make it too, in Germany e.g. "Dimaval" from Heyl, but that's not the main chelator for lead, so just FYI.
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By the way, even though I studied CS and business, I have a significant background in physiology, anatomy, chemistry/org.chem./bio.chem., statistics (and reading papers) so that I am very comfortable with all the basics, even if I know nothing about diseases (interest in how things work, but not in becoming a doctor who knows how things break). So I'm not prone to fall for wild esoterics but actually read papers and talked to scientists such as my researcher doctor.
What I find remarkable is that the problems of heavy metals are actually all known, for many decades. After all, the world abolished lead from fuel (except for small airplanes fuel) - quite a feat when they have trouble to come together to do anything else about subjectively more obvious issues. Also, you can find on many websites from organizations such as the NIH and other reputable sources, the medicinal threshold for lead is zero. Same for mercury. When I took a course about water treatment (Tufts University) the topic "lead" was a whole quarter of the course. A medical professor explained at length why zero is the desirable level.
So, people - doctors, officials, "the Internet" (commenters) - have no problem recognizing it as a problem. However, whenever there actually is a patient going to a doctor, heavy metals never play a role, unless it is one of the exceedingly rare acute poisonings, mostly industrial, and when you ask about it you are a nut case, and not all those ignoring well-established science.