Wood ash is fine to amend with in small amounts but can also drop ph. That's all that "cooking" the soil does: Give time for the microbes and fungi to consume the organic material you just added and to normalize ph. The more active the soil is the faster the process. Typically takes 30 days.
A sterile soil with a bunch of stuff added to it will not be very active. Some stuff like say manure or guano has bacteria in it but nothing like the multitudes that fresh ewc has. You need active compost to make soil active and introduce microbes to colonize in it. Fungi is also present in compost but adding granular mycorrhizae at transplant guarantees you have the symbiotic varieties you want attached to the roots.