I thought about this some more and came up with an additional reason.
The Church promotes indulgences partly to help the persons doing the indulgenced work to grow in grace by doing the indulgenced activities, which include things like giving money to the poor, praying Rosaries, adoring the Blessed Sacrament, reading Scripture, etc. The indulgence is a motivational tool to get us to do something that is helpful to us spiritually. If I read Scripture for a half hour, I not only grow spiritually and receive God’s grace for reading the Scripture, but I also can earn the special “bonus gift” of an indulgence. I can keep this indulgence for myself, or give this indulgence to a deceased soul in Purgatory because that soul is unable to get an indulgence for themself or do anything for themself at all.
However, another person walking around on earth is able to earn his or her own indulgence, and the Church would prefer that they did so because one purpose of indulgences is to motivate people to do various spiritual practices. If living person A is doing these practices and giving all the benefits to living person B, then person B has no motivation to bother reading Scripture, giving to the poor, etc. which is not what the Church wants to see happen. They want to see living person B earning his own indulgences.
Living person A is of course free to pray for living person B, or have a Mass said for B, so it’s not like we can do nothing for other living persons, even though we can’t give them indulgences.