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The algorithm $E'(m)=E'(k,m)=E(0^n,m)$ is defined with a hard-coded key, thus the key is part of the algorithm definition of $E'$. Because of Kerckhoff's principle we generally assume the attacker to know our algorithm definition. Because of this, the attacker can just try decrypting the challenge ciphertexts of the eavesdropper security game himself (or ...
If we encrypt $m_1$, and send it to the server, can the server "somehow" find $E(m_1)$ and remove it? Nope; FHE allows a server that knows $E(m_1)$ and $E(m_2)$ to produce a ciphertext which is a representation of the value $E(m_1 \odot m_2)$ (for pretty much arbitrary functions $\odot$); what it doesn't allow a server to do is determine whether \$m_1 = ...