I transplant runners... I've read where you should rotate where you grow strawberries. I've ripped mine out several times and moved them, never buying new plants. No issues here.

My strawberries that I planted in 2010 are still doing splendidly. Actually I bought one single plant. Last year, I didn't even water it at all and I thought it was dead. Nope. This year I decided to give it some tender loving care for the heck of it, keeping it watered and fertilized. Also I uprooted it and planted it in new soil, actually in a pot, not knowing what I really had planned for it. Now the darn thing is sending out new runners like you wouldn't believe and has spread out at least ten-fold. So my take is that yes, you can easily breathe new life into old plants if you wish. See linked video for details. I've since added several more pots as it wants to keep sending more runners, and they all grow with ease. Next year, I'll plant all these in the ground in a nice sized strawberry patch.

Here is a link that might be useful: Propagation of Strawberries

So I think I understand that provided you keep track of your runners, that each individual plant should produce well about three years, but I am still not sure what to do now that disease is an issue. Would you transplant runners to a fresh clean bed since they are coming from a parent plant that has leaf spot? Surely some of the spores will come with the soil if not the plant itself, no?

Rejuvenating the patch by means of heavy pruning will probably do the trick. The plants are usually good producers until the tap root gets woody af which point you should pull that plant and replace it with a new runner.

Strawberries are pretty hardy, you can basically mow them down and they will grow back strong. That is actually how some people rejuvenate their patches... Just don't do that with ever bearers or day neutral types, only June bearers.

Do you see the disease before or after they bear? We have plants that have been in the same spot for several years that continue to bear well. They tend to look diseased after harvest, but always green before. I suspect it may be senescence not disease, but either way, they continue to bear very good berries. We've also been taking runners and planting them in other areas, and they continue to do well as well, although no better then the mother plants, so I'm not sure how valid the three year rule is... in my situation anyway. We almost never water them as they seem very drought tolerant, even in our dry sandy soil when everything else is wilting. We have several varieties as well. Stawberries seem to be very robust in well drained sandy soil.

I should note that as maintenance free the strawbs seem to be, we do yearly pruning and mowing down. That may be why the plants are still producing like champs after several years.

It only appeared after I refreshed the bed. The irony is last year I did not refresh as vigorously, and I saw no signs of spotting. Once the new growth came in, I was worried about the heat/drought, and I made the mistake of putting down an irrigation system consisting of low sprinklers.
I'm going to give this bed one more year, but I'll just have to evaluate how they look before I decide to transplant the runners into a new bed next year.