A very large H-bomb can be made almost arbitrarily clean, if you are willing to compromise (a lot!) on other qualities. It would be possible to use bombs for excavation work and have the resulting added radiation to be hard to distinguish from the natural background.

Not that this makes using bombs for excavation a good idea...

If there are no bad after effects, then why is it not a good idea? We already use explosives for all the things listed. Stopping hurricanes sounds pretty useful about now.

Firstly, stopping hurricanes with bombs will not actually work and will instead make them worse.

Secondly, because clean nukes need to be so huge that they are simply too dangerous for their utility. Think at minimum "more powerful than all conventional explosives ever produced" level huge.

While doing things like channeling to the Qattara depression sounds promising, the actual practical problems of operating with true city-busters are just too great. There are plenty of industrial accidents caused by explosives every year, you don't want to live in a world where a really bad accident can wipe out a city.

There's still fallout and there's the problem of nuclear proliferation.

There's a reason the world looks in askance when nations start enriching uranium, even if it can be used for good.

Is it just politics that makes this a bad idea or are there other engineering problems?

With all due respect, there is a discete element to fallout. The radiation on aggregate might be at a particular (and assuming you're correct, I don't know) low/background level, but that won't help you in in inhale a small Pu particle.

There is not going to be any Pu left over after you initiate fusion on 10x the mass of the primary worth of DT right next to it. The amount of fast neutrons that releases will fission every last heavy atom found in the core. This is precisely why arbitrarily large bombs can be made arbitrarily clean. As you increase the proportion of energy produced by fusion, you reduce not just the proportional fallout in relation to the size of the bomb, but the total amount of fallout.

The second problem is of course those very energetic neutrons -- to make them not activate enough material to produce even worse fallout, you need to wrap the bomb in something that absorbs them and produces something benign.

Also, just as a minor nit that always bothers me: Plutonium is never your problem with fallout. If a bomb goes off or a reactor burns, the other things that are released will always be many orders of magnitude more dangerous than the plutonium. No-one has ever died of, and no-one will ever die of cancer because of plutonium released in a nuclear explosion, reactor leak or a reactor failure; simply because to have enough exposure to plutonium for that to be likely, you will have developed a thousand other cancers caused by other, more dangerous isotopes that you would also have been exposed to.

(People might have died cancers they got from working on plutonium for bomb, rtg or other use, but that was only possible because the plutonium was first separated from the more dangerous isotopes and well purified.)

What are the kinds of compromises that would need to be made?

Mostly, they need to be very powerful, more powerful than is practical for most use, and they need to be physically very large.

Can you provide more detail on this? I have never heard such a thing.

I think it's likely, having just harnessed the power of the atom after growing up w/o electricity, Teller assumed the problem of fallout was a simpler one to solve than what he'd just worked on. We see the standstill, where radiation was unresolvable... but wouldn't he see it as a smaller hill after he just climbed a mountain?