Asking as a total novice on the topic. I want to act as a proponent for nuclear energy but it’s difficult to know what’s propaganda and what’s legitimate.
(In either case, the recycling process also separates out all sorts of really nasty, short-half-life fission daughter isotopes & such. Which could make it pretty unpopular, even if the mere chemical aspects of the process were non-toxic. Which they aren't...)
Short half-life (days) products are very radioactive and dangerous, but because they are quickly depleted, they are manageable.
The worst are medium half-life (decades) products. Short enough to be dangerously radioactive, and long enough to be a long term problem. These are the ones that make Chernobyl inhospitable.
So an article equating long half life with danger isn't trying really hard to be accurate and precise.
To put it another way, "half-life of infinity" = "not radioactive at all".
Now, there are other factors involved (type of radiation emitted, what fission products appear further down the decay chain, etc.), but none of them are made worse by a long half-life per se.
It's a scare term, pure and simple.