They both periodically send "proof-of-works" that prove that they are working
This step could mean a significant overhead while performing the POW using ELC because they have to do additional proof of works beyond the one you are interested in. On the other side of it, the ELC network could be huge.
I agree, this comes at a overhead. The reason why we have to do this here is that we have to measure how much everyone has contributed.
People can submit work where they are interested in many many output, but they as well may submit a work which is impossible to be solved (like factor a 256bit number). In both cases we want even and constant payout for other people's contributed computation power, and so they must periodically give some kind of verifiable feedback.
But in order to reduce the overhead, we can make this feedback relatively rare. We would have to think about what (security)implications the timespan between two proof-of-works has.