^^^^ Only if it's done as a hysterectomy. Snipping the Fallopian tubes is done via laparoscopy.
Still a general, still an overnight stay, and still an invasive surgical procedure in the abdomen. You get a small infection from this you are in a VERY different place from a man who gets a small infection from a vasectomy. People treat this stuff like it's no drama because we generally don't kill people in surgery anymore. However, that doesn't reflect the recovery and issues in recovery at all. Abdominal surgery is a big deal.
I've been told it's possible to combine this with a C-section, so that would require no extra abdominal surgery in many cases.
Of course surgery is always something to be careful about, but I don't think sterilization of the male partner is something that can ever replace sterilization of the female herself - if the woman is the one who doesn't want or medically shouldn't go through pregnancy she should have the option for permanent anticonception, not the male she is currently with.
I just checked the official guidelines in my country and they state that there is no randomized study available that compares the safety of female sterilization to other methods of birth control, that sterilization is generally safe and the general anesthetic is the biggest risk factor.
A C section is MASSIVE surgery. You could combine anything with it. Hell, lop a leg off and it's not that much more serious! It wasn't that long ago that C sections killed 100% of patients. Even today, although obviously the survival rate is nearly 100%, the post op is not a lot of fun for many people. Infections are common, scarring and strictures are common, internal adhesions that cause ongoing pain are very common, hormonal issues can be common depending on the surgery. None of this stuff is walk in/walk out, regardless of anecdotal cases.
Surgical sterilisation IS generally safe. The surgical issues are rare. The post op recovery issues are not. Which do you think hospitals use to let you know the safety of their procedures?
Chemical sterilisation of women is an option that men don't have. That's the point you seem to be willingly missing. You can have an subdermal implant, you can take a pill, half a dozen other options. I'm not even including shitty options like IUDs. Any of those is ALWAYS preferable to surgery. Of course the option to get pregnant or not should be in the hands of women. No one is disagreeing with that. But there better be a damn good reason why you would subject a patient to that risk when there are other very good options available. And if surgery comes down to being the only realistic option, let's see if the male partner will take one for the team first. Women don't have to have surgery to manage birth control. So why the hell would you? (Not including particular medical issues that really do make surgery the best option)