Behind your reasoning is the assumption that if you don't chop the forrest you are in the wrong spot. That's not very brilliant reasoning.
In a lot of situations chopping a forrest leaves you with 3 'things'. Food for example Or 2 food and one hammer in a lot of other situations. A lumbermill provides 3 'things'. So it is really a matter of what needs the city the most.
No, you're wrong.
A city needs 1 thing - Food. Period. Once you grow to a large enough size, you have everything you need.
The reason that people are telling you that your reasoning is wrong is because it is. Your cities need to grow and build basic infrastructure. The faster they do both, the faster they reach their full potential. Lets use an example of a forest, and you're building a granary.
Leaving a forest and building your granary 5 turns later means that you're 5 turns behind the guy who built his first. That's 5 turns of +2 food +1 food for every granary resource you are working in the city. That's an extra population right there. You're not working the forest on T1, so your idea that you get consistent hammers is wrong. You would never work a 1f1H tile. When you're starting out, you want to be working 3F tiles, or 2F1H tiles. This spreads to 4F tiles and 3F1H tiles after Civil service. Even if you improve that forest to 1F2H, you're not growing working it. 1 food doesn't support the citizen that's working it.
Lets say you and I have a forested grassland and a forested plains on a river. If you put lumber mills on both of those, you would be working a pair of 1F2H tiles, for a total yield of 2F4H. If I instead chopped them both, and put farms there, I would be working a 4F and a 3F1H tile. That's 7F1H, which is more "things" than your dual lumber mills, in addition to the fact that I got free hammers for chopping the forest AND I have enough food to support 3.5 citizens, not 1 citizen, growing my city EVEN MORE so that I can work more tiles. That extra yield means my city now has another citizen and a half who are working more "things" than yours are.
My logic is solid. Yours is looking only at the term of total yield, which is wrong. Food is FAR more important than hammers early, and if you prioritize food, you will find your cities will be able to work EVERYTHING later on. You will have massive cities where you work all your farms, all your mines and all your specialists, whereas yours will be stuck working a couple of lumber mills and being very low population.