Displacing something against the gravitational field, gains it potential energy. Moving something against the nature requires work. If the electric field of a negative source charge 'Q' points inward, then by definition, it would require some work to displace a negative test charge 'q' against 'Q'. According to the Coulomb's law, like charges repel and unlike attract. Two negative charges are like, and they are supposed to be repelling each other and no work is required to keep them apart. I think the electric field of a negative charge pointing inward is nothing more than a surmise. What does a theoretical physicist think?
However you choose to define the fields, you will run into the issue that there are two species of charge, positive and negative, whereas with gravity, there is only one species of mass.
One consequence of this is that the gravitational interaction is always attractive between any two massive bodies, whereas the electrostatic interaction is either repulsive or attractive depending on the relative sign of the two charges. This distinction between the electrostatic interaction and the gravitational interaction is probably ultimately what is tripping up your intuition about work and its relation to moving charges around in electric fields.
In particular, recall that the electric potential $\Phi$ is defined in terms of the electric field as $$ \mathbf E =-\nabla\Phi $$ One consequence of this is (the details of which I will leave to a textbook on EM), is that positive charges generically move from regions of higher electric potential to regions of lower electric potential (they "roll down" potential hills) whereas negative charges move from regions of lower electric potential to regions of higher electric potential (they "roll up" potential hills).
I agree that at first glance, this is all very counterintuitive since you're used to massive balls rolling down hills etc., but this is because intuition obtained from the gravitational interaction isn't quite sufficient to gain intuition for electrostatics. Also, it seems you may be worried that there is some violation of conservation of energy going on in electrostatics because no work is required to move like charges apart, I assure you that this is not the case.
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