I've been reading the answers to related questions, but I'm still confused about the variable of the length of brewing time. I've heard that when using a Moka pot, one should turn off the flame immediately upon hearing the very first gurgle sound because if it brews a little bit longer (before the pot starts spurting coffee) the resulting liquid coffee can taste more bitter. I've also read that one of the benefits of the Aeropress is that the "brewing" time is so minimal that the coffee tastes better and is "cleaner". If less brewing time=less bitter,then does coffee made with a french press inherently taste more bitter than other methods? (Maybe people that use the french press prefer more bitterness?) I also read a post elsewhere from a person who says she bought a single-cup coffee filter holder with three holes at the bottom rather than the typical one hole, and that the coffee tasted better because the water drained more quickly. Does the faster the coffee passes through a filter make it less bitter? If so, I wonder why the three hole filter holder isn't more sold more frequently. The art of coffee brewing can be so mysterious ;/
As discussed further in this question about bitterness, degree of extraction is one significant factor in bitterness in your cup. There are a couple of other links to coffee bitterness in that question also.
Keeping other major factors constant (e.g., same temperature of water, same beans), time and grind level are the easiest to modify to reduce bitterness. That is: shorter time and coarser grind will (generally) result in less bitter coffee. While you certainly can't expect to get a reasonable brew from barely chopped beans in 15 seconds, small changes in time and grind level can have a noticeable impact on outcomes, especially for immersion-style brewing methods like French press, Aeropress, pour-over, and to a lesser extent stovetop methods like percolator or moka.
Specifics vary by brewing method. For instance, some bitter-tasting compounds are filtered out by paper filters, so paper-filter pour-over will generally be less bitter than French press, for example. Also, your average paper filter will allow brewed coffee to filter more quickly with a coarser grind (finer grinds will more quickly clog the filter) so you might have to vary your pouring method to keep the total brewing time constant.
You might enjoy trying to vary one parameter at a time: trying coarser- or finer-grind coffee, then with a shorter or longer period of time, in both cases trying to vary only one parameter at a time.
The comment on the Moka pot I believe has a bit more complex answer than what you are indicating. Moka Pots are almost more similar to an espresso brewing method than a french press or a pour over system. Water very near boiling is forced through a coffee puck at some level of increased pressure which brews your coffee. Just looking at the mechanical system that is being used, it appears to me that if you were to leave the pot on the stove just slightly longer than intended, your coffee would brew with a higher temp water, at higher pressure. Both of these can cause bitterness. I think the comment stating that increased time on the stove = more bitter coffee doesn't have to do with brew times, but rather, has to do with water temp and pressure.
A lot of what you are asking comes down to the chemistry of the molecules in the coffee grounds. When water is applied to the coffee grounds, the first compounds to extract are citric and malic acids followed by more complex (and less soluble compounds). The compounds which contribute heavy body tend to have high atomic weight, and are not very soluble.
Presuming your grind, water temp, ratios etc are all the same, then the longer you steep the coffee grounds in water, the more body will be contributed to the final extraction, while a very fast extraction will contribute lots of brightness and acidity.
French press is on the far end of the spectrum for brew times, so the resultant coffee has lots of body, but the lighter flavors tend to be muddled, and the acidity seems blunted. An additional factor on the french press is that the filters are all in all pretty poor, so you end up with lots of fines in your cup of coffee. Fines are not dissolved solids, and don't really contribute to anything but mouth feel. They might lend a bit of body, but really, they leave that nasty last sip of grounds in the bottom of your cup.
Typically drip / pour over are a bit more balanced. They filter fines, and brew time (ie water contact with grounds, not necessarily total brew time) is shorter, which contributes to a 'cleaner' cup of coffee where the acid and finer notes can be easily perceived, and the body doesn't completely take over.
Many variables, all have effects. Find the ones that result in coffee you like, and don't get too concerned with it.
I have been using a french press exclusively for several months now. I don't particularly like bitter coffee - but I also don't use water nearly as hot as is common with other methods (or pretty much required for them to move water at all, in some cases) - sometimes I let it cold-brew overnight, and otherwise I stick to water at a temperature I can drink without being scalded after it brews for 4-5 minutes (without adding any cold water or milk products.)
I also use a mild/medium roast, not any of the darker roasts. With moderate temperature water, another 10 minutes does not make a big difference over my usual 5. I'm not slaved to using a timer and sometimes I get to doing other things while the coffee steeps.
If you want a harsh, bitter brew, use boiling water in a french press and leave it a long time. Using a dark/burnt roast could only help. I'll pass ;-)