Guess what, there is an entire wikipedia page devoted to your questions. A history of malaria vaccines.

> do what we have always done with bacteria: target cell membrane, burst the cell and rinse and repeat?

Not quite correct. Majority of antibacterial drugs leave the bacteria intact. Instead they interefere with the reproduction rate. Either slowling it enough for the immune system to clear, or metaphorically putting a condom on to stop reproduction.

Those routes don't work for plasmodium. Unlike bactera which are very uniform, plasmodium are very diverse within their own culture. You can spot bacteria onto a plate and it usually grows one big blob; do that for plasmodium and you find lots of little and big blobs. Any route that targets plasmodium reproduction will fail because there is a huge evolutionary pressure to develop drug resistance. All those mixed genetics clusters in the same infection will compete and at least one will be drug resistant.

A human who is infected with malaria and recovers will be immune to the disease. However, they aren't immune to the parasite. They can be reinfected and asymptomatic, spreading it to other people. A person can even be permanently infected.