I'll be quick, since I'm in a bit of a hurry.

No, you absolutely do not. Logicists hold that all truths within any system of logic can be deduced from logical propositions within it. Godel proved that this is fallacious. Neither appeals to external evidence physical or otherwise. — Barry Etheridge

Logicism is a very broad umbrella, so it wouldn't surprise me if some logicists did hold the views you're attributing to them. Nevertheless, that's not how we generally characterize, e.g., Frege's logicism or contemporary logicism (defended mainly by Crispin Wright and Bob Hale). The main point is not that one about truths, but about concepts, namely that every mathematical concept is reducible to a logical concept. For instance, Frege thought (correctly!) that the concept of "number" was suitably reducible to the concept of "class", and the latter was supposedly logical. Obviously, this implies that a good number of theorems that we consider as characteristic of numbers will need to follow from this new characterization, which is why Frege spent so much time trying to show that this indeed happens. As it turns out, he was also correct in this: Crispin Wright, George Boolos, Richard Heck, and others have shown that the axioms for second-order Peano Arithmetic are actually derivable from a single principle known as Hume's Principle (it should actually be called "Frege's Principle", but Frege modestly attributed this principle to Hume, and the name stuck), a principle that can be taken as an implicit definition of the concept of number, a remarkable fact that is known as Frege's Theorem in the literature. If this is enough to vindicate logicism depends on the logicality of Hume's Principle, and this is a highly disputed matter. Regardless, this has nothing to do with Gödel's theorems.

It requires physical evidence to prove that classical mathematics are a subtype of Intuitionistic mathematics that are more fully expressed using a metaphoric emotional-logic, hence, Godel's Theorem can merely be considered to be begging the question and demonstrating that classical mathematics are incomplete. That would make it official that classical logic describes about a quarter of everything observable really well and another quarter to a more limited extent. — wuliheron

I don't understand what you say when you say that classical mathematics is a subtype of intuitionistic mathematics. Given that there are theorems which can be proved in an intuitionistic setting, but not in a classical setting, and vice-versa, they should be disjunct "types", that is, there is no relation of inclusion among them. Can you clarify?

Also, you said that Gödel's theorem is "begging the question"; begging the question against what? What is it assuming that shouldn't be assumed?