One frequent argument against the safety of the synthetic sweetener aspartame is that it can be hydrolyzed, with one of the hydrolysis products being methanol, which is known to be toxic. Of course, usually not much aspartame is added in food anyway to yield a sufficient amount of methanol or its oxidation product formaldehyde, but this does beg the question: why wasn't, say, the ethyl ester chosen to be developed as an artificial sweetener? Is there a structure-activity relationship between the alcohol used to esterify the dipeptide and the sweetness of the product?
This question raises more important issues than just the technical "why methyl ester," so I'll address those too.
The easiest explanation for their focus on the methyl ester is that the ethyl ester just isn't nearly as sweet. This report says it is approximately 10x less sweet (see Table VI on page 2689 and the entry 'Asp-Phe-OEt' with "++" being defined on pg. 2684). It's an easy economic choice between the two, assuming the only difference is sweetness/g, but you're also worried about the perceived safety concerns of the methyl derivative.
A toxicology class I once took can mostly be summed up by a cliche phrase, "the dose makes the poison." Even too much of a good thing is by definition a bad thing, for example too much oxygen, water or even the sun can lead to death. We also say that too little of any compound won't cause measurable harm.
It's correct to say that aspartame will be metabolized into amino acids and methanol. Methanol itself is toxic in high doses, in the same manner ethanol is, by acting as a CNS depressant. This acute poisoning can cause respiratory failure or other medical emergencies such as kidney failure. Ethanol poisoning occurs in microorganisms too: it has been proposed that some yeast produce ethanol as a means of eliminating competition. This would have been potentially deadly for a prehistoric mammal too, if it were not for enzymes which metabolize ethanol, called alcohol dehydrogenase. ADH lead to biochemical pathways that allow the safe removal of ethanol, but methanol is a different case. ADH instead make formaldehyde, which is then converted to formic acid by ALDH. Formic acid, in large quantities, is used in the animal kingdom as a weapon, most notably in bee venom and from the bites of a fire ant. In low quantities it is fairly benign. Formaldehyde is just a nasty chemical. Both are produced in the body as a natural response to methanol and it's easy to see how too much of either couldn't possibly be desirable. The body also has ways to eliminate these toxic metabolites assuming you don't overwhelm these systems. Hence why a sufficiently small amount of methanol won't make you blind.
Researchers have measured the amount of these undesirable metabolites in the blood after having people, like me, ingest nearly 1.8 g of aspartame in one sitting. Then they looked at urine and blood concentrations and they couldn't detect any formic acid in the blood but they could clearly see it in their urine for, up to, 8 hours after a dose. This evidence suggests that acute methanol poisoning isn't going to happen after a ~ 200 lbs person ingests 1.8 g of aspartame. Let's stop for a second and put this in terms of something we can imagine.
I don't have any reliable numbers, but let's just say that for a 12 oz diet pop there are 200 mg of aspartame. That means for every liter of pop, there are approximately 564 mg of aspartame. That means 3 liters of pop would get me to 1.8 g of aspartame. That means I would need to drink more than 3 L in one sitting to overwhelm the excretion systems. This suggests an acute poisoning isn't very plausible.
There is, however, long-term toxicity to consider as well. This is a much more difficult question to probe. "Sure, there is no acute toxicity, but what happens in the long-term with a low dose?" For the answer to this question we have to turn to toxicologists who have not reported long-term issues. They are looking too. The general consensus is that there is no long-term harm known for the general population. It is hard to quantitate these things, because to perfectly do the experiments necessary would require controlling a bunch of parameters that human experimentation laws prevent, let alone funding constraints. So, researchers make the best with what they have available and using these tools they cannot find any measurable problems with this particular compound in the long-term for the general population. Their analyses are limited by uncertainties in their measurements and this is just the general nature of science.
Tomorrow the majority of scientists in this area of interest could be proven wrong by a very careful experiment, but over forty years of research hasn't shown this chemical to be dangerous at the doses we're likely to be exposed to for any duration of time. We have to make decisions off of the best evidence we have available. I think it's okay for people to apply some level of skepticism to these things, because in science we're continually correcting our accepted knowledgebase. Once in a while the general community is proven wrong about something long thought to be true and everybody benefits when that is the case, but we can't say the general community is wrong without substantial evidence. It is for this reason that we have the FDA to evaluate the best available evidence and via a network of reliable strangers, we can enjoy a very cheap/sweet drink that is low in calories. Frankly, I am more worried about the amount of sugar I take in than I'll ever worry about the amount of aspartame I consume -- we have already good evidence that sugar, with a dose too small to cause acute poisoning, is probably killing hundreds of thousands of people due to long-term exposure.
The most likely reason to use methanol instead of ethanol is cost.  Ethanol is roughly twice as expensive as methanol.
Although both aspartame and the ethanol analog can be produced with reasonably high yield, the ethanol analog has not yet been produced commercially, so it would be even more expensive to develop industrial scale production of the ethanol analog.
Finally, to address the toxicity part of the question, aspartame is not at all toxic. It is most stable at lower pH (~4.3) which is in the range of carbonated beverages. So methanol is not generally produced while the food is in storage, and the body gets rid of methanol as it is produced.
One more note, aspartame can break down via heating. This is why food that is intended to be baked is not usually sweetened with aspartame (or it is mixed with saccharine). Although the fear here is not toxicity so much as cake that isn't sweet!
Aspartame's sweetening properties were discovered by accident. the G.D. Searle company was trying to develop drugs to treat ulcers in the 1960s when one of the scientists accidentally contaminated his finger and licked it. Knowing how many versions of molecules get tested in drug development, my guess is that an ethyl ester version was made, and probably failed due to improper taste or some other property. Aspartame is already "Sparingly soluble" in water, I assume the ethyl ester would be even less soluble, making it less useful for sweetening drinks. However, the amount of aspartame that would have to be dissolved is so small I don't know if it would make a difference. It's probably taste related.
As for toxicity concerns, it takes a lot of methanol to cause toxicity, wikipedia claims 10mL can cause permanent blindness, and 30mL can be fatal, but the amount of aspartame a human would need to ingest to produce 10mL worth of methanol would be ridiculous.