Just some napkin math:

(Suppose, as it hasn't been tested that the) SpaceX Starship can carry up to 100 tons of material to LEO. It costs 100 million USD per launch. (11,000 tons / 100 tons) * 100 million USD = 11 billion dollars. The US Department of Energy (DOE) has a budget of 149 billion, so under these pessimistic conditions it would cost around 7% of the budget. That's not good, but its not crazy either.

Elon Musk has stated that he thinks the cost of launching Starship will go down to 10 million per launch in the next 2-3 years (not sure if thats realistic). On the other hand, we probably don't want to unload a bunch of nuclear material in LEO, but much further away from Earth.

Just watched the Kurzgesagt video. They make the point that more radioactive material is put into the world by burning coal, which I think is a valid point. They use the Falcon 9 exclusively to calculate the cost of launching, and they never indicate in the video that the cost of putting 1 kg of material into space might go down. Falcon Heavy can already launch (reusable) 63.8 tons to LEO at a cost of 97 million, which is around 1500USD/kg instead of the 4000 they cite.

They also strangely compare it to the cost of the fissile material, which doesn't make any sense as the majority of the cost for a nuclear reactor is in safety precautions, not in the actual production of nuclear fuel.

In their sources they're a little less forward:

Electricity from nuclear reactors is produced for about $70 per MWh, which is 7 cents per kWh. The fuel costs represent just 0.46 cents per kWh or 6.5% of the total cost.
If we had to get rid of nuclear waste by putting it on rockets, causing fuel costs to rise 3.4 times to 1.57 cents per kWh, the total cost would be increased to 8.1 cents per kWh. This means the total cost becomes 16%
higher.

In the video they state "In 2021, we saw a record 135 launches into space. If we repurposed each of those rockets and filled them all with nuclear waste, the total amount that could be lifted into a Low Earth Orbit, which is the closest orbit above the atmosphere, is nearly 800 tonnes. "

And then they extrapolate this out to... forever, I guess? Either way there were 178 successful launches in 2022. There have already been 123 (successful) launches in 2023. Regardless it seems reasonable to expect this number to go up and the average tonnage per rocket to go up. It just seems strange to make this video that is very explicitly only about 2021 and to have the economics change so much even in 2 years. I would've preferred if they stated in the video that it doesn't make sense today, but it could in 5 or 50 years, and made more arguments in favor of storing it deep underground.

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The biggest risk (as another commenter mentioned) is the massive risk to the environment by trying to shoot it into space. If we really needed to get rid of it, we could bury it under a mountain for 100 years, dig it back up, and shoot it into space when the cost is much lower, and its much safer to launch.

Or just never launch it at all. I don't really buy the idea that burying nuclear waste in remote areas is dangerous. Critics say that the problem is that you have to keep waste contained continuously (say for the next 10,000 years) but I'm near certain we will have some way of truly getting rid of it in the next 500 years. Whether that's blasting it into space or reprocessing or something else entirely.