We use half life because, quote simply, that's how the laws of thermodynamics work in chemistry. I know that's kind of an unsatisfying answer, but answering why it works that way is kind of like asking why radioactive elements decay in a half-life fashion.

When reactions generate a half life, it becomes very difficult to say precisely when the reaction will stop proceeding entirely. Additionally, even if you knew precisely how long it would take, a large percentage of that time would be taken by the tail end of the reaction, where blood levels of the drug are so low they are ineffective.

In short, we use half life because it gives the amount of time that the concentration of the drug is in your system is at the level we need it to be for it to be beneficial and do its thing.