They usually only remove half and do the frozen section when they are unsure of what it is. In my case, for example, my FNAs had come out inconclusive, but the nodule was growing, so I had half of my thyroid out in one surgery. Unfortunately, they couldn't tell what it was with the frozen section, so they left it at that. A week later, I find out that, yes, it was cancer and I had to go back in again a month later. But I was mentally prepared for that.
Had they known from the get-go, it is generally accepted protocol to take out the entire thyroid in one surgery. I think that you're going to get that opinion with probably at least 9 out of 10 doctors you talk to, not only because it reduces risk of recurrence, but also because it makes it much easier to monitor after treatment. Also, you can't have radioactive iodine treatment (RAI) unless you have a total thyroidectomy. I think also that this has been the way that thyroid cancer has been treated for more than 50 years and it works. Especially with Papillary. So the answer just becomes to do what has always been done, regardless of the stage. It's also a relatively easy treatment compared to other cancers, so they don't see a reason not to do it.
Yes, a lot of people have issues with taking the pills for the rest of their life, but most doctors would want to keep you at a suppressive TSH level, at least for the first 5-10 years (although I don't know what the protocol is for only having a partial thyroidectomy), which would require you taking thyroid hormones anyway, albeit at a smaller dose and you wouldn't be dependent on them 100%.
I think it's something you need to decide for yourself, but don't make the decision based solely on not wanting to take the pills. Consider both sides, the risks included, and make an informed decision. Ask what protocols would be involved if you did only have a partial thyroidectomy, such as yearly ultrasounds to make sure there are no new nodules on the other side (although some doctors order them even with a total, at least for the first few years to make sure everything's okay).
I am all for you getting a second opinion. Nothing wrong with that, that's your right and I think more people should probably do that in many cases (especially with some of the stories I've heard on this board, you wouldn't believe!). But don't just dismiss their opinion. Challenge it. Ask them why that's the protocol and if it has to apply in your case. Write down a list of questions you have and make sure you understand the answers.
I hope this helps. I'm not trying to push you guys into anything, even though it really sounds like I'm an advocate for the traditional treatment. Just make sure you know everything to expect with both options and then decide.
Best of luck to both of you. Please re-post with what happens or if you have any more questions.