Nodal analysis with ideal operational amplifier
Your proposed equivalent circuit is not equivalent because it does not provide any mechanism to force \$v_+ = v_-\$ and it doesn't provide any mechanism to inject current at \$v_o\$.

If you want solve an op-amp circuit by nodal analysis, you can replace the op-amp with a very high-gain amplifier or VCVS.

For pen and paper analysis it's usually simpler to not strictly use nodal analysis but use the negative feedback equations (\$v_+=v_-\$, \$i_+=0\$, and \$i_- = 0\$) instead (after you've confirmed that negative feedback is correctly established).