No, this is a fabulous strategy, although it needs some modification. It isn't clear to me what you see as the downside as it beats a conventional start in almost every way. In fact, I predict that some variant of this will be a key in successful MP online strategies in the future, as well as for GOTM winners. There is simply no other strategy that can really compete with it.
I am currently in a game using a modified version of this strategy, and from the beginning, my score has been 2-3 times that of any of the other civs. My early tech path was bronzeworking ->monotheism (for the religion) -> pottery (for cottages) ->alphabet (to trade techs).
Here are the important takehome messages.
First, the chopping of lumber is key to productivity at certain times. Later in the game when cities are well developed, the 30 productivity from forest chopping adds little. But early on when cities are producing just one or 2 hammers per turn, lumber chopping can really jump start your civ. This can be used for many different things -- settlers, workers, wonders, etc. But it makes the most sense to chop the trees early as this can put you way, way ahead at a time when other players are spending 17-25 idle, stagnant turns waiting to pump out a lone settler.
Second, it makes the most sense to use tree chopping for workers and settlers, NOT for other units or buildings. The reason is simple: while building military units or buildings, your cities continue to grow. While building workers and settlers, they do not grow at all. Other things being equal, it is a tremendous advantage to shorten the build time of settlers and workers, because this minimizes the time that your city is stagnating and not growing.
Third, I employ this strategy on a rolling basis. I didn't stop at 4: I went up to 15 cities this way on my large pangaea map with great success. Of course, you have to diversity. I pumped settlers and workers out of some cities by chopping trees, and after the trees were gone, I developed the economy by building cottages and mines, developing resources, and later, building watermills. While I was chopping trees in one city, another was building military units, another was building missionaries, and yet another was building city improvements. You get the idea. By doing this, I was able to keep research constantly at 90-100% and had a very solid economy.
Fourth, I reserved forests around a couple cities for wonder building. Forest chopping is a fabulous way for non-industrious civs to get key wonders very quickly. Say you want to build a wonder with speed doubled with stone, and have stone (while other players don't) due to your rapid expansion (+100% build speed). You also switch to organized religion (+25% build speed: it DOES work on wonders, not just regular buildings, I saw from my current game). This gives you a +125% hammer bonus. For harvesting one square of forest you get not 30 hammers toward your wonder, but 67 hammers! Harvest a few squares of forest, and of course no one can keep up with you in wonder building. For the next wonder, go to another city you have plopped down in another area of forest. I was able to get every wonder that I wanted. I had to be selective, but got stonehenge -- hanging gardens -- notre dame -- chichen itza -- etc. etc. on a large map with 8 other civilizations! This is such a good technique that honestly I am happier with non-industrious civs (expansive/financial is how I played my current game -- the boost allows fast granaries and makes up for any health benefit of all the tree chopping), and will likely find little use for the industrious trait in the future.
Finally, the concept that taking down forests will harm productivity is nonsense. Forests offer little food, often even enough for sustenance, and so cities that rely on forests for productivity often stagnate. Upgraded watermills give +2 production, as well as economic and food benefits of flatland. The forest productivity is easily replaceable. Chopping the trees has helped me to vastly increase, not decrease, my city productivity by allowing much more time for growth, bypassing large amounts of stagnant time waiting for worker/settler production without this strategy.
Through this whole game, all my settlers have had military escorts, and my borders have been well defended. Those who have declared war on me have been quickly crushed. I have maintained a constant flow of gold, high (90-100% research -- now way ahead of the other civs by the mid middle ages), and city development, and have gotten all the wonders I wanted.
My thanks to Alexfrog for bringing the forest harvesting idea to my attention.
Best wishes,
Falconhurst