Caveat: Leather armor is a debated topic amongst both professional and amateur historians. Its effectiveness, cost to construct, how common it was, everything about leather armor is in doubt. This answer is based on my own interpretation of sources, and I have included several links at the bottom from differing perspectives and as primers to start you off on your own rabbit hole of discovery.
I mainly want to know whether lower thickness would decrease armour
Yes. Thicker armor always makes for better protection. All armor isn't thick because of added weight, cost, and flexibility.
If yes, could this gap be bypassed with the woven patterns?
Partially. Weaving leather will usually make it more resistant to piercing, for example from a sword thrust or arrow. It might change its protection value against blunt force damage. Leather armor is already quite resistant to slicing, and weaving will have negligible impact on this. Weaving can improve flexibility, but only to a point; more on that later.
Tl;dr Simplified solution: use woven leather for the chest piece (cuirass). Solid leather pieces for the lower arms (vambraces), thighs (cuisse), shins (greaves), and helmet. Cloth fabric for the stomach, groin, and shoulders (either gambeson or another form of cloth armor, like linothorax). Pair it all with a wooden shield. This gives you decent protection that is cheaper and lighter than metal, while also easier to repair. It is also a low-tech solution, enabling your society to use it long before innovations from the high middle ages.
Explanations: What is leather armor?
Leather armor is not supple, soft, or comfortable. Leather had to be hardened to offer protection. This was usually done by boiling, but can be accomplished in several ways. This link is a great one to start with for understanding the boiling process, if you're interested.
Hardened leather is still more flexible than metal plate. Leather can be hardened to become plate- or wood-like, but becomes brittle. This especially hard leather can be used in some ways, such as lamellar, with the expectation that it will need replacement pieces after most battles. For other uses, hardened leather is not made to be that stiff, so it will have some flexibility and give, but it will be like the flexibility of a leather shoe sole, not the sides of a leather boot and definitely not like a pair of driving gloves.
Leather armor does not breathe well; wearing it is a sweaty business. It is lighter than metal, especially if the metal is iron or bronze and not a lighter alloy like steel.
How effective is leather armor?
This depends entirely on the type of armor. Lamellar, brigandine, simple boiled leather cuirass, each type has different properties.
As depicted in many games and fiction, leather armor looks something like this. This is a cuirass. Assuming proper construction (and not a costume), it will not allow the wearer to bend completely in all directions, but it will be fairly light-weight. It will offer some protection against blunt force, though repeated hits will cause stress and eventually break the armor, and padding under the leather will greatly reduce the injuries the wearer takes from such attacks. Piercing attacks made by a sword like this will have a difficult time penetrating, and slashing attacks will likewise not penetrate often.
Extremely sharp swords will easily pierce or slice leather armor. Most battlefield weapons are not "extremely" sharp; keeping that edge is nearly impossible. As soon as you sheath the blade or actually hit anything, the edge will dull. In a pre-steel society, and especially pre-iron, nothing in battle will have such a sharp edge.
Weapons designed for armor penetration, like a medieval rapier, will pierce. Spears will be your most dangerous foe- they allow the wielder to put enough force behind a thrust to penetrate your leather armor, particularly if the spear tip is long and thin. Arrows and javelins will also pierce, provided the hit is not glancing and has sufficient force behind it (hunting bows, with for example a 20lb draw, won't get the job done, though a war bow with a 70lb draw will). Crossbows are unlikely at 600BC tech levels, but if used, they would penetrate.
What about studded leather armor?
There are arguments over whether studded leather armor ever existed historically, or what "studded armor" even is. In games, it is often used to signify leather armor "upgraded" to better protect you. This just isn't the case; metal studs are not going to stop penetration or help against blunt force, and any impact on a slashing attack will be negligible.
Studded leather armor could be a description of pieces of leather held together with metal studs, hence the metal holds the armor together instead of reinforcing it. This would be somewhat similar to lamellar armor, though less flexible. Most historical examples of studded leather were likely brigandine; the amount of metal used means it can't be considered leather armor for our discussion here.
But I asked about woven leather
Weaving leather prior to the hardening process is certainly possible. This should make it easier to obtain material, as strips of leather left over from other projects could be used (to a degree). The complexity is increased, but not so much as to be impossible to make.
What does the weave actually accomplish? It allows the leather to be slightly more flexible, as the pieces can move and shift. The hardening process means it won't be a night-and-day improvement, but it will be noticeable. Unfortunately, this extra flexibility introduces friction, which will wear the leather out as it rubs against itself, reducing the lifetime of the armor. A tighter weave reduces flexibility and friction both.
More importantly, it will improve your protection from piercing strikes. This does not mean the woven leather will be immune to piercing strikes, merely a little better. That said, the weave introduces seams that can be exploited by dedicated piercing weapons, such as the previously mentioned rapier. A tighter weave will be more resistant to this.
There might be a beneficial increase to protection against blunt damage. Speculatively speaking, it could allow enough flex to reduce the stress such blows cause, thus helping the armor last longer under repeated strikes. It would need to be tested before hanging your hat on it, but is likely reasonable enough to pass mustard for fictional works.
Are there other alternatives to woven leather that fulfill your needs?
Flax gambesons could be an alternative. The cost of making these compared to leather armor is debated, but cloth armor in some form was used by nearly every society in history. Layers of cloth quilted or glued together are much better protection than one might imagine: modern kevlar armor is essentially a type of cloth armor, with layers of synthetic fibers put together.
As previously described in the tl;dr above, a combination of leather, woven leather, and cloth armor would probably be the best possible approach your ancient culture could use. The different pieces could be replaced independently, allowing for easier manufacture and field repairs, while providing the benefits of the differing types. It would probably look pretty awesome, too, something like this but with a woven leather chest piece.
Additional links and references: 1, 2, 3, 4, 5, 6, 7.