Hm... probably not. Potentially, this could easily expose the server to resource exhaustion attack. In order to support this, most likely the server has to remember the search result so far in memory (or somewhere), until the logical end is reached. This is costly, and could be used for attacking.

Consider this design difficulty: the server cannot guess whether you want to continue search, so it has to remember everything as long as there is still possible results left unreported. An obvious question would be when should the server release the resources, and forget your historical result. This is almost an unresolvable question. There is simply no way the system can safely guess your itention, maybe use some sort of timeout, which is not ideal.

The current way is nearly the best. (I am not a developer of this site, just a user ;-)