There is no real way to 'source' this because it is totally a modern perception. In the past, bowmen were not reviled as weaklings. As you already know, to draw a bow requires strength and to do this repeatedly during a battle requires great reserves of endurance. Fighting as an archer took a lot of skill and years of practice as well - these were not just sickly farm boys dragged into the army - they were professionals.

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The perception of archers as weak probably comes from games and movies. Games tend to be all about 'balance' so if an archer is an expert at shooting things from far away then the designers will balance that by making him weak, which is not the case. For a famous example of how this is untrue, look at the Battle of Agincourt, where English archers fought in the frontlines against French knights and won. Besides a bow, archers would often have a mallet, dagger, axe or some other weapon to be used at close range.

Movies tend to make archers smaller because it is a way to show a slight-framed person as a powerful warrior. To add to this, modern movies love placing women in medieval-action scenarios and while most audiences won't find it believable for a woman to don heavy mail and go slug it out in a melee, they do find it plausible for her to do the Hollywood Ninja Archer routine, which at least in cinema goers' eyes does not require strength.

You can find something similar with Asian martial arts. Nowadays in a movie if a woman/child/teen/small man is going to kick some ass in a movie, they are likely going to surprise their overconfident enemy by performing some Asian-derived martial arts moves. Now, these movies rarely even get this right but it is at least meant to mimic it, probably out of some racist view that Asians are weaker physically than other races but can achieve victory in combat through speed and ingenuity rather than pure power.

This one is tough to answer as this isn't so much a historical question as much as it is an evaluation of the development of tropes in the fantasy genre. In ancient literature, the use of powerful bows was always associated with strength and physical prowess, from Odysseus in Homer's epics to Yi the Archer in Chinese legends. Up to the Napoleonic era in early 1800s, writers still pushed for the inclusion of (non-existent) archers alongside muskets, though their skill and strength was often overstated, given that hundreds of years had passed since bows were used for warfare. Pretty much no piece of historical literature or training manual would have even mentioned that archers were weak.

I think we'd really have to look at pioneers of the modern fantasy genre, such as Tolkien, to see where our perception and association of archers being agile fighters comes from, especially with bows being connected to forests and trees, which are the abode of Elves, noble and agile in combat; in contrast to the flawed realms of Men and the industrious Dwarves. The proliferation of tabletop and roleplaying games, such as Warhammer and Dungeons & Dragons, would have cemented the "balanced" perception of the effectiveness of archers as ranged-but-weak fighters.

Edit:

In mentioning Homer, I may have overlooked a contradiction.

Great heroes such as Odysseus and Herakles were said to be capable of drawing very strong bows, and of course they were no weaklings. However, Greek literature also frowns upon the use of the bow in single combat, described as the "mute weapon of the worthless man", referring to the cowardly use of the bow compared to spear or sword, the story being used in reference to Paris in the Trojan War.

So in Ancient Greece, the bow would have been used by both the strong and the weak, depending on the context and whether the individuals are being portrayed as heroes or villains.

Source:

Todd Davis, Archery in Archaic Greece