To continue and elaborate the idea hinted in @Carey Gregory's comment ("I suspect the answer is that if a drug isn't safe and effective for both 100 and 400 pound patients, it's not going to be an OTC drug"), the answer lies in the safety margin between efficacy and toxicity of the drug over a wide range of dosages.
According to the paper "CRITERIA FOR SWITCHING NEW MOLECULAR ENTITIES FROM PRESCRIPTION TO OVER-THE-COUNTER" (1990) by Paula Botstein, MD, Deputy Director, Medical Affairs, Food and Drug Administration (at the time):
"To be a serious contender for OTC marketing, [the drug] should have
- The drug should have a relatively favorable adverse reaction profile. It should
have relatively low toxicity, as do the antihistamines already marketed OTC.
- The drug's expected use should be appropriate for an OTC drug; for instance, allergic rhinitis is appropriate but chronic urticaria is not.
- The drug should not have properties which make it undesirable for OTC use, eg, a doctor's oversight needed for monitoring during therapy special cautions or restrictions on use or other limiting characteristics."
Focusing on the first property of safety, the safety margin mentioned earlier is known among healthcare professionals as the "therapeutic range" or "therapeutic window" or "therapeutic index" (see figure).
The figure depicts the percent of response to the drug among the study population as a function of weight-adjusted dose (mg/kg body weight, note that this is a semi-logarithmic plot). There are two curves: the leftmost one shows therapeutic effect, which slowly rises with increasing doses until it reaches saturation (100%), meaning that beyond this dose, there is no further benefit from taking this drug. The rightmost curve shows toxic effect, meaning that beyond it the response to the drug is toxic (i.e. adverse effects).
The therapeutic window/range/index is the distance between those two curves. The farther they are from each other, or the larger the dose range between them, the wider is the therapeutic window and the safer is the medicine, since the dosage required for eliciting adverse effects is higher, and vice versa (when the curves are closer to each other the drug is said to be a "narrow therapeutic index" (NTI) drug, and treatment should be monitored carefully to avoid adverse effects).
This concept can explain how a single dosage can be suitable for people from very different weight groups. Suppose, for example, that the therapeutic index depicted in the figure belongs to acetaminophen. Now, if you give a 500 mg acetaminophen tablet to an adult weighing 50 kg and to another adult weighing 100 kg, the former would receive a dose 10 mg/kg of acetaminophen, while the latter would receive a dose of 5 mg/kg - both dosages are found well within the therapeutic index of acetaminophen, because it is wide enough to accommodate both. Thus the same dose can be appropriate for two people with very different weights.
Of course, there are other considerations for the "Rx-to-OTC switch", as the process of determining which drugs can be sold as OTC drugs is called, e.g., is the drug addictive, how dangerous is an accidental (or non-accidental) overdose, does the treatment require a doctor's monitoring for dosage adjustment, etc.
Also, here is a very similar question and answer.