For believing Latter-day Saints, universalism speaks to a legitimate problem and a legitimate yearning.
The problem universalism speaks to is as follows: exaltation is the product of choices in the direction of eternal life (God’s life) that are made by souls who see their options with clarity. In mortality, many of us — to some extent all of us — have trouble seeing our choices with clarity. Some of the reasons for this (sin, rebellion) are within our control, but other reasons for our inability to see clearly (culture, trauma, neurological wiring, lack of opportunity) are to some extent not within our control. So after this life, there will be periods of time where people come to see reality with the clarity that has not been possible in mortality. Universalists are confident that everyone who sees with this clarity will choose eternal life, or that God will somehow unilaterally impose eternal life on everyone, regardless of their choices.
The legitimate yearning that universalism answers is the desire to be with loved ones for eternity. Latter-day Saint universalists see in the statement “families can be together forever” not a statement of possibility, but a statement of divine intention. They view any possibility of eternal separation from loved ones as being contrary to God’s plan, and view the eternal gathering of our Heavenly Parents’ children as being an ideal that is fully within our Heavenly Parents’ power to achieve.
There is a degree of legitimate truth in these universalist ideas, and we would do well to honestly acknowledge that.