In gene transfer by microinjection, the gene of interest is injected into the nucleus of host cell without using integrase enzyme. So why does HIV need that enzyme? Why they not just place their DNA into the nucleus and let it stay there?
During gene transfer by microinjection the DNA has at least some fragments of identical DNA sequence to the host genome. Usually DNA that is used for microinjection will have flanks on either side identical in sequence to two nearly situated, sometimes adjacent blocks of DNA sequence. This allows homologous recombination to occur which engages proteins already present in the host cell: https://en.wikipedia.org/wiki/Homologous_recombination
In case of HIV, there are no fragments of identical sequence flanking the genome of the virus, so no homologous recombination can occur, therefore integrase is necessary. Furthermore, integrase inserts viral genome into random parts of the host genome unlike homologous recombination, which would always localize that viral genome into the same spot each time.
Now, this is my wild guess, as I couldn't really find any evidence in the literature but I'd say it gives the virus certain advantage. Site-specific insertion could be easily inhibited in the course of evolution by mutations within the host genome and eventually the virus would be eliminated. This is not the case with random allocation, as no mutations in the host genome will ever substantially change the probability of integration.
Here is a good review on the process of DNA integration regarding HIV: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3385939/
Let me know if this helps, confront your opinions with what I think.