I know that in regards to today’s bows, leather wrapped limbs often indicate that the limbs are made of fiberglass and the wrapping is there to somewhat disguise that fact, but did archers of old ever wrap their limbs and, if so, what’s was the purpose of it?
Wrapping comes with a cost: it makes the limbs heavier, and therefore reduces the efficiency of the bow. Thus, unwrapped limbs are often preferred. But some traditional bows had/have wrapped limbs. Wrapping can be for four purposes:
Reinforcement of the bow.
Weather-proofing.
Ornament/display.
Magic.
Reinforcement. Sometimes limbs are wrapped to reinforce the bows. For self-bows, wrapping can help prevent splintering, or stop splintering/splitting from getting worse. For composite bows, wrapping can help stop de-lamination and can reinforce joints in the structure (e.g., where limbs join handle, where ears/siyahs are attached to the limbs). In both cases, wrapping can help hold the pieces of the bow together in case of major failure, making it safer in such an event.
Reinforcement can be restricted to the parts of the bow that are most likely to fail, or can cover much or all of the limbs. An example of a minimally-wrapped bow:
This composite bow (19th century, Bashkir) has wrappings at the string bridges, and the base of the ears/siyahs, and on one side of the grip (perhaps the other side of the grip was once wrapped, too). More extensive wrapping can be seen on this bamboo bow:
where there are multiple wrappings along the limbs, and a fairly large section of wrapping on the lower limb. On this self-bow from Tanzania:
much of each limb is wrapped. More common on African bows is to have wrapping restricted to the limb tips, either the very tip, or the limb just below the nocks - these are the most important locations to reinforce to prevent splitting. The wrapping can be cord or leather thong (and I have seen one example where the limb tips were wrapped with iron, in a manner similar to wrapping of spear butts and hafts around tangs). African bows with limbs completely wrapped in leather thong are the main example of leather-wrapped limbs I know of.
Japanese bows (of laminated bamboo or bamboo and wood) often feature wrapping along the length of the bow. These are often discrete bands of wrapping along the limbs, and can be few or many. Sometimes, they are completely wrapped. The wrapping is often rattan, or silk thread (sometimes, silk ribbon). One example with closely spaced bands of wrapping:
Composite bows are often wrapped, but invisibly. The entire limb is often wrapped with silk thread, and lacquered (which waterproofs the bow and also covers the wrapping string).
Other bows are wrapped with thread, too, such as South American cotton-wrapped bows (e.g., bow 201 in Grayson 2007). Silk and cotton thread provide a lightweight but strong wrapping, and are an efficient way to reinforce the entire length of the limbs.
Weather-proofing
Composite bows are often wrapped in birch bark. The main function of this is weather-proofing rather than reinforcement. The bows are often painted or lacquered as well. Weather-proofing is important for composite bows, since the organic glues that are used are susceptible to moisture. Self-bows can be efficiently (i.e., with less weight) weather-proofed with wax/oil, so wrapping for weather-proofing is usually not done.
Ornament/display and Magic. It can be difficult to distinguish between these, and indeed a wrapping can serve both purposes (and can also help reinforce the bow - for example, the cotton-wrapped bow referred to above (bow 201 in Grayson 2007) is decoratively wrapped in alternating black and white bands). In many cases, the wrapping serves no physically functional purpose, and can sometimes add sufficient weight to impair efficiency. Some examples:
A Somali bow with many bands of giraffe hair, with each band supposedly represented a killed elephant: https://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=597234&partId=1
A bow (South-East Asian?) with cloth wraps on the limbs: https://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=553000&partId=1
A Melanesian bow with the limbs covered in plant fibre woven in an ornamental pattern: https://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=504811&partId=1
An African bow with lizard skin wrapped around one limb.
Reference:
Charles E. Grayson, Mary French, and Michael J. O’Brien, Traditional archery from six continents: the Charles E. Grayson Collection, University of Missouri Press, 2007.