Agreed, it is not a guarantee, but for most children, even very sensitive children, it does work.
If it's not getting ANY better at all after say 2 weeks of consistent repetition (and long enough to actually be worth doing e.g. at least 20 mins like someone else suggested), or the child is getting more and more distressed and is not able to put their attention to anything else at all, then it might be something to get concerned about, but it can take children a longer time to settle than we would ideally like. I live in Germany and there's a very slow and gentle settling in process to nursery, both my youngest took around 6 months to "officially" complete it, but they are both very settled and happy now and love it there. I definitely had points within that process where I wanted to just pull them out and cancel the whole thing because I was afraid it would never work and I was just distressing them for no reason, but I'm really really glad that I didn't. They get so much out of it now.
There are markers of progress that you'll see in this process, that I learned from the German system.
A child who is distressed because their parent is not there will typically cry much of the time, they won't move from one spot, often a place where they last saw their parent (like a door/window) they won't engage with any toys/activities, they don't want to eat or drink, and they won't try to engage with the new caregiver.
But you can start to see that they are feeling more comfortable when they cry less of the time, even if they are crying on and off for some or all of the session. This will eventually reduce into crying only at the point of separation and later not at all.
Another marker of feeling more comfortable is when they start to move around the room and engage in different activities. Often at first, they will switch between doing something for a short time, then going back to the "safe place" and crying a little and then going back to explore the toy again or something else.
And most importantly they tend to move from ignoring the caregiver, to starting to initiate interaction with them (e.g. looking or gesturing at them, talking to them, showing/handing toys to them, pretending to feed them etc and other kinds of play), and then the most important milestone of all is when they start to accept comfort from the caregiver, so e.g. the caregiver will offer them a hug and they accept, or they will want to sit on the caregiver's lap. At this point if they are still at the stage of wanting to stay in one place, the caregiver's lap will often become their "safe place" and they will do their explore-return-explore to here (often without crying as they are finding it comforting to be with the caregiver). Then they will start to accept more caregiving activities - often in the order of food, nappies, last of all sleep support. (Under the German system, they keep the parent nearby and don't force the child to have their nappy changed by the caregiver if they are not yet comfortable, they would just end the session if a nappy change was needed).
It's not just a case of saying "ignore the crying it will go away, it doesn't matter, it doesn't mean anything, it's just manipulative." (Whatever people say when they don't understand the process). It's a process that they can go through and understand that they can be sad when mum/dad leaves, but they can also have fun and that they are safe and nothing bad happens and then mum/dad comes back and they are OK. It might not be an easy process to go through, but that's OK. We don't have to protect children from ever having to do anything hard.
There are probably children for whom this won't work, but to write it off so early does not make sense, when the OP has already seen some signs that it's likely to progress well (can leave child with dad, child happily engages with other adults when mum is present, crying solidly changing to crying on and off).