This is an interesting idea but I think it doesn't work. Say for example that another species starts 200 million light years away and is spreading a colonization wave at .5 which is a pretty extreme value. Then one should have at least 400 million years to notice that. And it is going to be pretty hard to do a fast colonization wave without some astronomically detectable signs. Reducing the colonization speed makes it less likely to be detected but increases the time span.
It seems no less plausible that what spreads outward at a sizable fraction of lightspeed is a wave of "terraforming" agents, altering all planets in the neighborhood into more suitable colony planets. Meanwhile colonization spreads at a rate roughly bounded by the ratio of reproduction rate to death rate, which might well be significantly slower than that.
That scenario would be enough to ensure that if an intelligent species evolves, it is necessarily far from any spreading interstellar empire (since otherwise the terraforming agents would have ...
If it's worth saying, but not worth its own post, even in Discussion, it goes here.