In our parish, the priest doesn’t pray for the people who are sick. We have a prayer chain and once in a while the priest will mention to pray for the people whose names are in the book. Again they said that took attention away from Christ in the Mass. The priest doesn’t even name the person who the Mass is for. I always thought that was why we had the Mass for them.

Hmmm.  I never met a priest who does not pray for the sick.  I suspect the priest may be concerned – and understandably so – about protecting the privacy of individuals.  On the one hand, it is very good that the parish community be concerned about the welfare of everyone in the parish, especially the sick and the shut-ins; on the other hand, there can be legitimate liability concerns when it comes to individual privacy.
 
As for “taking attention away from Christ at Mass,” I am not sure how it would do that. Jesus often sought out the sick and marginalized as especially worthy of his time and attention. Anything which promotes the corporal or spiritual works of mercy (visit the sick, give drink to the thirsty, feed the hungry, etc.) by necessity leads us to Jesus Christ.
 
Finally, if the priest does not mention the intention of the Mass, the Mass is still valid, and I am confident that the priest has that intention in mind during the Mass. Nevertheless, when the celebrant mentions the intention of the Mass, he is actually inviting everyone at Mass to join in that prayer of petition.  I think that’s a good thing!
 
If you would like to know why your priest does these things, you could ask him.