I think you have two different issues in your workflow.
The first one has been addressed in the comments. Data Merge. It also can be done inside Illustrator. You can link, not only names, emails and phones, but also graphic resources, such as photos.
But regarding the lag using 25 logos, probably there is a couple of them that are not optimized.
If you have some time, see if you have unnecessary shapes or elements on the logos.
Ideally, a logo should be a vector element, but perhaps you have a really big raster image as a logo of thousands of pixels when you need a really smaller size; or you have a thousand vector elements making a texture; things like that.
So, I would do a check on each one. I would also see the weight of the file itself. For 25 logos for a business card the file should not weigh more than an Mb, a few at most.
Also, a logo should not have blending modes or effects that need to be redrawn each time you move an element.
Another thing to consider, (I am not sure if it is worth it for a business card template) is to learn the difference between embedded elements and linked ones.