That rare exception is if they believe the players are on a level that leads them to believe that the best possible outcome for the recording would be to ditch the click in favor of a 'live' click.
That's the thing - there are many more players that think they are in this category, than are actually in it. If you're assessing it subjectively while you're playing it, it's really not a valuable opinion. If you can record yourself for the length of a tune playing steadily and you don't have tempo drift, that's a decent indicator, but even that isn't enough.
Let's say a song is doing a slow boil build up to a dramatic louder section, with no particular tempo change written into the song. The dynamic level drops slightly for a few bars to add contrast just before it goes to the hardest hitting section, which is a perfectly good, proven device.
A player that is caught up in the song emotionally, and is already harnessed by the initial downturn before the crescendo, is probably going to have a 'tell' about the forthcoming section. It may be a change in their part, or it may be a tendency to speed up at that point. Is this good ? It depends who you ask. The player may think so - they are feeling the emotion of being in that situation, and you can hear it in their playing, so on some level that is "good", because there is some expression. But is that the best thing that can be done in that moment ?
Think of it like you're a film director. Which would you rather have, right before an unexpected critical moment - an actor's performance where the words start to come faster and more excited, tipping off the audience that something is about to happen, or one that is laid back and in the moment, unaware of the future just as the audience is ? If you're going for the latter, in music, a click is the most practical tool to encourage it. The better way is for the performer to know that they tend to speed up or get louder in the more exciting parts, and to be able to compensate for it so that they can have the full range of options available to them - but not everyone can do this. There are people that play for decades that never figure it out. It requires awareness of your own subjectivity and a willingness to override your feelings and observations to get a desired result. Sometimes individual dynamics and tempo changes come at the expense of group dynamics and tempo. Those individuals usually are not good at seeing this.
As another example, I sometimes like it when a rock song speeds up a little in the chorus, that's fine. And it should be a little louder. But when it gets back to the verse or bridge, it should shift gears decisively right back into the original tempo, so you can feel the group as a whole establishing that contrast. That's a cool effect and one of the more difficult types of interplay.
There is a school of thought that feeling the band slow down halfway through that following verse is good. I didn't go to that school. Basically you're hearing that the players themselves became aware of the change only after they were well into the verse and it felt too fast (which it was). When players say they want that sort of thing, it sounds like an excuse to me. If you buy into that story, it appeals to the meme of "this is done by real people, not by computers, man." For me, there is always plenty of human variability and imperfection involved if real musical instruments are involved.