All compensated flying is "commercial" The 1)- 4) path you described is the usual route, because it works bestest, cheapest, and fastest in most cases.
There are a couple problems skipping the regionals. You normally do the regional to get the 1000+ Turbine PIC that is usually required by the Major airlines.
1) You could try to go from CFI to corporate/135. This could be easy if you know someone who owns or operates a turbine aircraft, or you could network while doing the CFI thing and try to hook up with a co-pilot job. You will need to be a first-class hustler to get this done. The downside is that the PIC jobs at this level are not usually stepping stones, so the incumbents are usually older and tend to stick around a long time. You could spend many years as a co-pilot before getting the opportunity to fly as PIC. Also in many cases, you will fly only a few hours a month. Be aware that the customers are high-end and demanding, and people skills are crucial...you must be comfortable hanging with the beautiful people in a social setting. Also be aware that pretty young women will generally have priority for co-pilot jobs...nothing personal, but that captain would rather spend the three day layover in Vail with her...anybody who says different doesn't know corporate!
2) The other big problem is that you generally have to know somebody to get a major airline job. If your uncle is the CEO or chief pilot at a major, you're set! Otherwise you need to meet some folks. Usually the way this works is after you get your PIC time at the regional, you will get recomendations from major airline pilots who flew with you when they were at the regional. If you skip the regional, you will need to come up with a plan B to meet people.
3) Majors prefer some 121 experience, but this is not an absolute deal-breaker.
You could also go from CFI to 135 freight, moving up to larger and eventually turbine airplanes over the years. You would have to research quality-of-life and upgrade times on this. I understand that these folks have no trouble getting major airline jobs.
There is one sure fire shortcut around the regionals: Fly military fixed wing.