[This answer is based on books 1 & 2 - and not any information from Death's End].
TLDR: My interpretation is that the droplet was never sent to kill Luo Ji. It was sent to interfere with Earth strategems based on "dark forest" principles, only reaching Earth centuries after Luo Ji was expected to have died alread.
But Luo Ji himself was confused about this, believing for years that the probes were in fact sent for him. The use of his confusion was a clever element of the book's written style and plot.
Consider when & why the droplet was sent...
Ye Wenjie shared her formation of the basic principles of cosmic sociology with Luo Ji. Even though at that time Luo Ji had no understanding of the consequences of those principles the Trisolarans recognized that information as a risk -- anticipating that the "dark forest" principle would eventually be understood by humans. (Learned this via the sophons of course.)
The Trisolarans set two both a short and long term plan in motion immediately after that event. This was the first time since establishing the sophon block that a serious, unexpected threat was identified in humanity. A threat which could manifest itself even given the sophon block, and long before their main fleet could reach Earth.
The short-range plan was to order the assassination of Luo Ji. An attempt was made on his life even before he was made a Wallfacer, and of course other attempts followed. The purpose of this policy was to try and eradicate the genesis of the dark forest concept before Luo Ji could even understand it or spread it. Assassination had the potential of preventing or at least significantly delaying humanity's realization of cosmic sociology and the development of strategies to counter Trisolarans based upon it.
(They could have tried to kill Ye as well, being the only other human known to Trisolaris to have this information. On the other hand they may have still regarded her as an ally).
The long-range plan was to get the 10 probes (including the droplet) to Earth as soon as possible. The 10 probes of the Trisolaran fleet began their acceleration towards earth as soon as Luo Ji learned of Ye's principles. Clearly this was not a part of the original invasion plan, and presumably the fleet must have made some compromises in order to make the probe's advance possible. This is evidence of the high level of risk the Trisolarans perceived.
However at that time the Trisolarans had no way of knowing what the conditions would be on Earth when the probes would arrive ~200 years later. In particular they couldn't know that Luo Ji would even still be alive at that time. So it seems clear that the intention of sending the probes was to give them the earliest possible opportunity to stave off any threat posed by humanity, including a strategy based on dark forest principles. But not to kill Luo Ji specifically.
But after the fleet's destruction as the droplet approached the inner solar system Luo Ji certainly thought it was after him. This is understandable for several reasons. Firstly he had already escaped Trisolaran assassination something like 8-10 times, so he was justifiably paranoid. Secondly, he knew for a fact that the droplet began accelerating as soon as he learned of the dark forest principles. (Learned from Ranier, I think?) (*)
But this fear on his part was an element of the excellent storytelling. The droplet was never sent to kill him, specifically, but instead it fulfilled its mission by (1) annihilating the human fleet, and (2) instituting the block of using the sun as an amplifier.
This seems to be rich irony. With the solar block in place the Trisolarans felt they had defeated Luo Ji's plan - that he could never send a second signal to the universe beaconing the locations of Earth & Trisolaris which so threatened them. And yet this was their utter downfall because they indeed should have eliminated him before he ultimately engaged his brilliant checkmate and nullified all their own strategems.
Luo Ji had in fact embodied the Wallfacer principle perfectly, being strategic yet unreadable, duplicitous, and misdirecting. The Trisolarans viewed him as a threat only as a conduit of the dark forest; but in actuality he himself was the threat.
(*) No one other than Luo Ji knew of the coincident timing of the probes' acceleration and his conversation with Ye.