The brittleness of obsidian (and other stone) is indeed a problem. A common way to avoid this is to use less brittle materials, such as bone, horn, antler, hardwood, bamboo, and stingray barbs. Head made of these material are less brittle than obsidian/stone, but are also less sharp. Despite being less sharp, they have been widely used for arrowheads and spearheads, and even continued to be used for arrowheads when iron was available (because they were cheaper than iron, and arrowheads can be lost).
Some examples:
Bone arrowheads, Siberia: https://research.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=538246&partId=1
Bone arrowheads, Iran: https://www.metmuseum.org/art/collection/search/324224
Bone arrowheads, Japan: https://www.metmuseum.org/art/collection/search/62273
Wooden (and other) spearheads, Australia: https://i.pinimg.com/564x/36/3f/ba/363fbab73909c2c940255a485765f8c0.jpg
Arrows with wooden heads, New Guinea: http://www.oriental-arms.co.il/photos.php?id=2148
Arrows with wooden heads, New Guinea: https://i.pinimg.com/564x/b9/a5/a8/b9a5a8ed35b4027b226b60a154a1c8bf.jpg
Brittleness is a lesser problem for arrows and throwing spears, compared with thrusting spears. A brittle head on a thrusting spear is a Bad Thing, and also a poorly-attached head. Bone and wood heads can be made with long tangs allow more secure mounting, providing two advantages over stone heads. The ultimate solution to secure attachment of heads for primitive spears is to simply sharpen the end of the haft, rather than using a separate head; this method was used for the oldest spears found (Paleolithic).
Iron is excellent for spear and arrow heads for three reasons:
It provides a sharper head than bone, horn, wood, etc. While an iron head won't be as sharp as a freshly-flaked obsidian or flint head, the edge will be more durable, and can be re-sharpened.
It provides a tough head. This also allows the head to be quite thin, which improves its cutting ability compared to relatively thick bone and wood heads.
A socket or long tang can be used to securely mount the head.
In addition, long iron foreshafts behind the head can be used for improved penetration, since the foreshaft can be thinner than the head is wide, as opposed to the arrow/spear shaft/haft which is usually wider than the head, resulting in more friction/drag reducing penetration. Very long foreshafts, such as used in the Roman pilum and some East African spears, can allow penetration far enough through a shield to hit the shield-user.
There is another reason why iron can be better: obsidian, or other suitable stone, is not always available locally. Obsidian is rare enough, and useful enough, to have been traded over long distances. Flint was also traded, and when surface sources had been exhausted, was one of the first things (or the first) sought via underground mining. Iron ore, on the other hand, is much more widely available, and bog iron and iron sand can be collected on the surface.
For a map of obsidian sources in the Mediterranean area, see figure 2 in Marie Orange, François-Xavier Le Bourdonnec, Anja Scheffers & Renaud Joannes-Boyau (2016), "Sourcing obsidian: a new optimized LA-ICP-MS protocol", STAR: Science & Technology of Archaeological Research 2:2, 192-202, http://dx.doi.org/10.1080/20548923.2016.1236516
