To the best of my knowledge, the Minhag by all on Simchas Torah is to first have Hakafos, and then complete the Torah. By a Siyum the general rule is to first complete the Siyum and then to start dancing. Why the difference between Simchas Torah and a regular Siyum? Is there a reason why one is different than the other? (looking for a sourced or a logical answer)
See p. 180 in this book. Hakafot on SImchat Torah was a relatively late addition done in the 16th century. One of the reasons mentioned in the source is that it is actually a supplication and extension of the prayer for rain. That is why we say Ana Hashem Hoshi'ah Na.
Reason #2 on that page notes that the Hakafot use one of the last phrases in Halel. I am surmising that because of this concept, the Hakafot were positioned closer to Halel.
Regardless, it does not appear that Hakafot have any connection to it being or resembling a siyum. If anything, perhaps, the Kiddush after you get the aliyah is closer to the siyum :-o