Perhaps that is the argument you were presented with. Normally, what I tend to hear is somewhere along these lines;
-the Earth looks flat
-the two options are it is flat, or it is round
-if it is flat, I have to assume what I'm seeing is accurate
-if it is round, I have to accept what I'm seeing and assume that other evidence (disconnected from this observation) proves the Earth is round
-the first hypothesis requires fewer assumptions, therefore Occam's Razor dictates that the Earth appearing flat is evidence the Earth is flat
Now, you may say that the first requires more assumptions, because you have to assume that the other evidence is false. However, the evaluation is simply whether the Earth appearing flat is evidence of the Earth being flat or not, that a hypothetical person with no knowledge but perfect reasoning capability is presented with only the fact that the Earth looks flat would reason that the Earth is flat. It only isn't evidence of that after taking into account other evidence, therefore Occam's Razor suggests it is evidence of the Earth being flat.
That logic makes sense to me. I don't know if I'm explaining it well, but that's how I've heard it.
First of all, Occam's Razor dictates nothing. It is a generality that is helpful in choosing between otherwise equal hypothetical situations.
Second, the way you frame the argument is an incorrect rationalization. We all agree the Earth appears generally flat at ground level. We all agree that this perception is what we would expect if the Earth were flat. Most everyone agrees that this perception is also what we would expect if the Earth were a very large sphere.
Earth appears flat = Flat Earth or Earth appears flat = Large Sphere Earth is the issue then, regardless of how you dress it up.
Rational people examine both statements and realize that there is insufficient data to draw a conclusion, and given only the statement "Earth appears flat" it could be either flat or a large sphere.
Irrational people decide that, though their single observation yields both results, the Earth is flat.
You claim there are more assumptions being made in the spherical Earth model because you have to assume other evidence is available. That's not an assumption because there is, in fact, heaps of other evidence available. Given only the appearance at ground level, the spherical Earth model makes no assumptions, and concludes that more information is necessary. The flat Earth model makes the only assumption, and concludes that the Earth is flat for no logical reason.