"...Can you make Aleppo soap with just normal Cold Process, Olive oil and Lye water and add Bay Laurel oil after trace? ..."
Yes, but why add the bay laurel oil after trace? That will not work to ensure the BL oil is the superfat -- the lye is still very active at trace and will saponify what it wants. If you specifically want the BL oil to be the superfat, you will need to hot process the soap and add the BL oil after the cook.
"...Do you think the Syrians went through that convoluted process just because they could measure the Lye percentages very accurately? ..."
Do you mean "...because they could ~not~ measure the Lye percentages..."?
If so, the answer is "yes" and "no". It's more complicated than that -- and here is a very simple but still complicated answer:
There are two types of "salting" as used in soap making.
Originally, only soft soaps could be made because the lye used for soap making was collected from ashes harvested from the burning of certain types of plants. The ashes contained mostly potassium salts, and these ashes were leached with water to dissolve the potassium salts and form a potassium lye.
The soaping fats were warmed in a large kettle, the lye solution was added, the mixture was stirred and boiled over a fire, more water and lye were added as needed, and the mixture was allowed to saponify. When the saponification was complete, common salt was sometimes added to the soap paste to force some of the potassium soap to convert into sodium soap. This paste was poured into containers for use. The salting step made the soft soap somewhat firmer and easier to handle and use, although the soap was still soft and paste-like. The sodium soaps in the paste were less soluble in water, so the soap would last somewhat longer. This type of soap was probably something like modern-day soft shaving soaps or "cream soap".
Later on in the early 1800s, soap was made with a sodium lye rather than a potassium lye. This changed the "normal" type of soap from a soft paste soap to a hard bar soap, similar to what we are familiar with today. This occurred when people figured out how to make sodium hydroxide (caustic soda) from soda ash and lime, so soap makers weren't limited to using the potassium lye from ashes.
This sodium soap was made pretty much as described above -- the difference comes after the soap is mostly or fully saponified. At that point, the soap is a hot, liquid mass partly mixed into and partly floating on a watery solution of glycerin, salt, impurities from the soap ingredients, and spent (used up) lye. Believe it or not, adding salt to this mixture will cause the soap to become almost completely insoluble in water, and the soap would form curds and float to the top of the soap kettle. This was called "graining" the soap. After a settling period, the water layer underneath the soap curds was drained off. This washing and draining procedure could be repeated several times, depending on what kind of soap the soapmaker wanted to make. The "grained" soap was poured into molds to cool and harden, and then cut into bars for sale.
Actually, commercially produced soap is still made today with a "boiled" method where lye is gradually added until the fats are just saponified. Only handcrafters make cold-process or hot-process soap using a superfat as "insurance" against the error in our calculations. In some respects, the boiled method has advantages, because no one can truly predict exactly how much lye is required to saponify fats and the boiled method allows for this fact.
Handcrafted soaping is actually more prone to error because the saponification values we use to create our recipes are basically estimates. These values may be very good estimates, but are still not precise measures of the fats actually in our soap pots. We just design recipes in a way that usually forces any error to be extra superfat, rather than extra lye.
Henry Gathmann, author of a commercial soapmaking text written in the late 1800s, talked a lot about the alternatives to the boiled soapmaking method. He described the cold process method and the half-boiled method (basically a hot process method). He felt these methods made soap that was less desirable in some ways than a fully boiled soap, because the impurities from the soaping fats were not removed. He wrote:
"...cold-made soap, therefore, contains—like half-boiled soap—all the impurities that may have been introduced with the stock, all the water used for making the lye, the foreign salts that may have been contained in the caustic, the glycerine formed during the formation of the soap, and also more or less of the raw materials in an uncombined state...."
He went on to explain the cold and half boiled methods were easier to do when making small batches, so small commercial (and home) soap makers were pretty much limited to using these methods versus the "full boiled" method. Gathmann went on to say:
"...The boiling of soap requires apparatus, labor, and time, which are too expensive to apply except for a fairly large batch, to say nothing of the practical impossibility of properly finishing a small batch of soap by boiling. In connection with this there is the further advantage that by the cold process a batch of soap can be turned out on very short notice, and certainly much more rapidly than by boiling...."
Probably too much info to answer such a short question ... but there ya go!