If you read about the history of the virus, supposedly in the past most people figured they'd get it (both variations) sooner or later and didn't particularly care. The stigma, especially of HSV2, allegedly started mid 20th century when acyclovir / zovirax came on the market and the company who created it needed a reason to push people to start using the medication. Nobody wanted to at the time, as HSV wasn't a big deal. However once you call people who have it "dirty" and make it shameful, the sales go up. Now you want to suppress outbreaks and potential spreading at all costs. Similar to AIDS in the 80s, in the US people won't even hug you if news leaks that you have HSV2.
Also amusing that, in the state of CA, receiving HSV (1 or 2) to the genitals from someone of wealth is a great source of compensation. Most of the money you'll receive from the civil case will not be for the medical treatment, it will be because of the "psychological damages and impact to lifestyle" from suddenly becoming a social pariah. Like you said, why this would be suable is not obvious. If 80% of the population has HSV1 and you transmit it to the genitals through oral sex, you're now a 2 year-long (statute of limitations) juicy target for a lawsuit. People of means have lost millions of dollars to this. Everybody having sex becomes a ticking timebomb of legal extortion. Better have your partners sign a consent form.