Stick frames can go up remarkably quick. The install also doesn't need any specialized tools that contractors don't already have, so speed and labor isn't much of an issue.

Getting long beams from a log tends, I think, to be harder than getting small dimensional lumber. The timber also tends to need to be higher quality, especially if you opt for a design where they are exposed (such as vaulted ceilings).

Mostly, it probably comes down to where you live and how available the materials are; if good timber is easy to get, there's probably more contractors available who are accustomed to working with it.

Likely because you need to hire a structural engineer to sign off on the project.

Building codes allow us to do this sort of thing without direct engineering input and most municipalities have not adopted a code that define safe practices for this sort of construction. Additionally, materials selection and availability is geared towards stick frame construction and you may not have everything you need from big box hardware stores or local supply houses.

You need a special contractor, and structural insulated panels for walls/floors. That said, the total man-hours of labor are probably less, and according to timber-framing contractors I've spoken with over the past three years, the price gap is now less than 10% more than traditional stick-built.