I don't even know where to start but I can tell you I am speaking from experience as I started off completing dumping and scrubbing my tank while now I have a cycled tank and would never ever change.
So quick answer is no, not at all!!! It is a million times easier and I'm a bit confused about where you have gotten this idea from, care to share?
In a cycled tanks you can still get diseases as not all diseases are caused by poor water quality but alot are from fin rot to swim bladder problems and many of these diseases will kill your fish. I don't think there is an preventative medication for the diseases you mentioned. I may be wrong but the medications for that fix the problem they don't work like an immunisation which will prevent you from getting the disease. So in other words they work like antibiotics
Also think about how much easier it would be for you instead of having to remove the fish from it's tank and completely scrub it and fill it up. Instead you just remove 10-20% of the water once a week, replace it with treated water and wash the filter material in used tank water every second week or so.
It is stressful on your fish to completely start anew every week or so. It is an incredible shock and this is why so many goldfish die after a few days. Goldfish can live to 10-20 years of age so can you imagine why these fish are dying at such a young age? There is no cycle that is built up so as the bad bacteria starts to grow they are attacked by their own waste instead of good bacteria feeding off it and therefore protecting the fish.

If your tank is cycled I think you'd see all this so I wonder if you have ever had a properly cycled tank.

I could go on and on but I know the others will cover everything else and probably explain it better

NO. It's 10 million times healthier and safer for your fish to live in a cycled tank since there are bacteria that take care of the ammonia the fish produce! Without that, they'd be sick and dying all the time.
But you need to test the water from time to time even in a cycled tank to make sure everything's okay! Like nitrate, that's very important as higher levels of nitrates go right on the fish's immune system and makes them susceptible to all kinds of bacterial, viral and parasitic infections. You need to keep the nitrates under control by doing regular water changes, about once a week, or once every second week in a bigger system. So NO, there shouldn't be diseases coming up all the time if you'll care about the water the fish live in, and if you'll quarantine any new fish before introducing them to the main system.

And don't treat them if they're not sick, absolutely not. That's another thing that will weaken their immune system.

Hi.  Thanks for clarifying this for me.  I agree with all of your comments, and am glad that you recommend a cycled tank, because it is much easier for me as well. 

I wrote the question because I used to have 2 goldfish in a 10-gallon tank that I dumped and scrubbed every month, and they lived 5-6 years with no disease and no problems.  (They died because of my neighbor's kid.)

I now have a properly cycled 10-gallon tank that I set up last March.  I test the water regularly and it's in great condition.  The only mistake that I made was that I put the fish in 2 days after setting it up, so they had to go thru the cycling process (sorry about that, now I know better.) 

One of my goldfish died 2 weeks ago from fin/tail rot.  And my other goldfish is really healthy and energetic, but yesterday I saw that one part of his tail looked torn, kind of like a cut from the end, towards the body.  I think this is tail rot, unless their tails are supposed to look like that?

Anyway, because of this, I started putting in 1/2 teaspoon of Melafix and 1/2 teaspoon of Pimafix last night.  I don't want to lose this fish too.  I plan on using these 2 medicines for the 7 days that the instructions say. 

Today I bought a new goldfish friend for my 1 goldfish, because if he is sick, the only stress I can see him having is that he was living alone.  He used to always swim next to my other fish that died.  So now I have 2 goldfish again in the 10-gallon tank.  They are still quite small; I know I will have to get a larger tank when they grow.

I guess I was wondering if there were more diseases existing in a well cycled tank, but from your response, it actually sounds healthier than dumping and scrubbing the whole thing. 

Hi Summer Time,

a rip in the fin doesn't mean fin rot all the time. If it is a rip about half cm or so it could be that he ripped it on sth.
Very good to treat with melafix and pimafix. the rip should heal pretty soon. I know, because our Humphrey, he's a sweet little bugger of a fish,  had this happen to him a couple of times. I'll find a pic, so you can see it. He also broke his dorsal fin somehow...?...no one knows how...

Cute picture! 

Yes, that's what my fish's tail looks like.  It's just a small tear and is hard to see because it doesn't always show with the way the tail lies. 

I did remove the carbon on the first day of the medicine, thanks.

That's a good idea. 

I wasn't sure about putting the new fish in.  But I think I read somewhere, that you should put in Melafix any time you get a new fish - so I thought I was taking care of both problems at one time.

I want to be prepared the next time I have a problem.  Do you treat your fish EVERY time he gets a tear?

What is the name of the medicine that you use for treating fin rot?

It depends how big the tear is, if it is small I don't treat, I watch it. If it isn't gone after a couple of days, then I start treating.
But it only happened twice to Humphrey, only.