I went the nuclear surgery route, IOW I swallowed a very large pill of radioactive iodine served up in a lead beaker while wearing a lead vest, kinda spooky. That basically killed my thyroid gland over the course of the next three months during which I went from hyper-thyroid to hypo-thyroid, started putting weight back on and got pretty sluggish. Since then I've taken synthroid every morning on a dose that's been very stable for at least the last ten years and really hasn't changed much in the nineteen years or so since I was diagnosed.
I got back on the bike fairly quickly once I got on the synthroid program but my racing was pretty lackluster for the next couple of seasons and stopped racing around '96 and pursued career and other sports for about a decade before renewing my racing license in '06 for another go at it. My body type had definitely changed post Graves disease and I'd put on a fair amount of upper body muscle and wasn't as lean as pre-disease.
I also started training with power meters not too long before I wrote that post you dug up from 2006 and am now working on my fifth season of training and racing with power data. I've steadily improved during that time, have had some great seasons and some average seasons but every one of the last five have been more successful than anything I managed pre-disease. My best 40K time trial back in the day with early clip on aero bars was 1:06 as a NorCal cat 3. I've ridden one or more sub hour 40Ks every one of the past five seasons and my best time is down to 56:15 so ten full minutes faster than what I managed when I was 20 years younger and had a normally functioning thyroid gland. My overall race results are also much better now with a handful of wins in crits, road races, tts, cross and even a few top GC spots in weekend stage racing in the past five seasons. I'm mostly racing masters now, but still hold my own in Cat 3 races and have managed a couple of podium spots against the younger cat 3s on good days.
I attribute most of the progress to changing how I trained, from having the power meter data for objective day in, day out feedback and from being more consistent in my year round training. But in terms of your question, it took me a while to figure things out but if anything my fitness and racing is much better now than it ever was before my thyroid gland troubles. So the disease alone once treated did not put an end to my cycling or racing and once under control wasn't that relevant. As I said my body type changed in subtle ways and I have to pay a lot more attention to weight management now or things get out of hand quickly but other than that I don't really think about it much.
-Dave