We have only recently figured out safe viral gene therapies (usually called lentiviral therapies)- the FDA approved the first two of them in 2022 [1]. I agree that to actually cure HIV, we'd need a system that can hunt down and correct the rogue DNA in our cells, but the technology is not yet there. First of all, to my knowledge, current lentiviral therapies aren't very good at targeting where in the genome they insert. That would be important to correct the implanted HIV sequence. However, our CRISPR DNA engineering systems are good at this. Understandably, there is work underway to combine the two [2], but academia tends to lead actual therapies by quite a bit. In the cases of both potential therapies, they wouldn't come close to screening / inserting into every potentially infected cell with our current technology. There's a lot of interest in improving this issue in pharma, though. Maybe it'll be possible some day, but not before a lot of work in the field.

I think about this specific question in the shower a lot, and am kind of stoked that somebody asked it.

[1]: https://asgct.org/publications/news/september-2022/eli-cel-second-lentiviral-vector-gene-therapy

[2]: https://pubmed.ncbi.nlm.nih.gov/34372494/