And what exactly prevents doing that with lamps of all types?
People's laziness to go few meters longer than into regular waste bin...
And a bit more technical differences:
Lead acid batteries all have very similar construction and composition, they could be "treated" all in the same way. And even then, only the lead is what is reclaimed for the use in new products. It is the high content of relatively pure lead, easy to extract (so it become the cheapest way to get the lead for the new battery making), what make the lead batteries so attractive for recycling.
And most important, the recycling infrastructure is set up and running already for decades...
Lamps are of too many varieties, each of them requiring separate plant in order to be efficient in reclaiming all valuable materials in usable quality.
The content of all the valuable components is only minute, while vast majority of mass being forms of glass and metal. While the toxic components tend to react with the surrounding materials, what make them more difficult to extract, so the recycling of old lamps is not the cheapest way to get material for new lamp making.