Great looking bed! Those plants look really healthy and growing under a cover can reduce foliar diseases.
The alkaline materials in wood ash are water soluble and can wash out (potters do that when making wood-ash based glazes) so it is primarily fresh wood ash that will push the pH up when you apply it to the garden. It also is difficult to make assumptions about the actual pH in your soils unless you have have them tested (state Ag programs used to do this for free or a low fee); to reduce incidence of scab there seems to be a consensus that you need to be below pH 5.2
The varietal effects make sense - diversifying into scab resistant varieties (and equally, away from scab prone ones as they raise the disease burden in your garden), and rotating where you are growing them year to year with non-solanaceous crops (no tomatoes, tomatillos, peppers or eggplant), should help. Well-composted materials are great to add nutrients and condition your soils, and clearly your plants are benefiting from that.