Yes, let them grow up and fill the space they are currently in.
Here is what I do for seedling removal. Typically I will start seeds in 4 or 6 inch pots (10/15cm)... to remove them, I try to pop out the whole group as one unit, to the extent the soil has compacted a little and they have rooted through it, holding it together. Usually I can push up on the bottom of the container and the soil will move up as a unit. Once the contents of the container are out on a plate (looking like before but minus the container) I will take the point of a chopstick and carve out soil around the edge that the plants haven't rooted into yet. Places around the edge where there are gaps, soil that is not being used by the plants.
Then if necessary I will push the chopstick in between the two biggest plants, or on the inside of the plant which will be easiest to start with (ie. the one closest to the edge). I put it in vertically down to the bottom, gradually, avoiding roots I might encounter, then gently move the top of the chopstick sideways in the right direction to encourage the soil to break along the line I have chosen. Lots of discretion here, you want minimum rippage and ideally each seedling will end up still rooted in a little ball of soil at the end. I do not bare-root the seedlings on purpose, but sometimes the soil falls away.
At the end I handle and pot up the seedlings mostly by grabbing and handling the root ball, not so much the plant itself. In the best case scenario, every seedling goes into a new pot with its roots undisturbed in original soil, and then gets filled in around the edges with new soil.
As for tools, tweezers work for spiny cacti. I like to use chopsticks. But mostly I try to use my hands if possible to handle the soil more than the plant, if that makes sense. One reason why I like young seedlings to root reasonably well into their starter pot before I even consider moving them along.