Thyroidectomy for most people is a valid option that should be open to the patient. It is surgery and carries risk to be sure. But depending on age, current health and such it may be a good choice for you.
I know my endo recommended RAI and I held off for awhile. I tried Antithyroid drugs for 18 months to see if my thyroid would calm down on its own. When I was backed off the ATDs my levels went through the roof again so I did the RAI. It took my thyroid about two years to die off and my levels to stabilize. During that two year period I like others went from hyper to hypo at the drop of a hat. It made it very hard for me.
In hindsight I would have done the thyroidectomy if I had it all to do over again. With the surgery you go hypo almost right away and you can go on replacement hormone and levels are easier to adjust because you are not producing any or negligible levels from what is left behind.
So talk to your doc, get your second opinion with the thyroid surgeon. If you decide on the surgery ask if they use a nerve sensor. It is placed on the vocal nerve and if they get close in the surgery it emits a beep. That way they know they are close to your vocal cord nerves and can stop there, thus greatly reducing the chance of vocal cord issues.
Let us know how you make out after you talk to the surgeon.