How to have edits of embeddable saved to search index?

Say we have an embeddable which we're editing through Embeddable Studio.. if I make a change to what it outputs on the page, it's visible instantly in the piece of content the embeddable is in...

All good so far.

However, what we've noticed is that the new content isn't in the search index.. it'll still be the old version that was first indexed when the actual piece of content was created;

I can see why this would be; the platform would have to go find all the instances of the embeddable in content & sent it all off to be reindexed.

Was this considered in the implementation of embeddables, and how do you suggest we address this problem? It feels like I might end up having to raise an Idea?