Abraham Cowley wrote that 'I never had any other desire so strong, and so like to covetousness, as ....that I might be master at last of a small house and large garden'. Should garden be large enough, small garden buildings are a welcome addition.
The ancestors of western ornamental garden buildings, in the west, are Greek temples in sacred groves. Each had a profound relationship with the landscape, because the place itself was sacred long before anything was built. When Roman emperors began to see themselves as Gods, their estates became 'sacred' groves the inclusion of small temples and grottoes was a natural development. The practice of building in gardens was revived with the renaissance and became a feature of English gardens designed to re-create the 'landscape of antiquity'.
Though much has changed, aesthetic garden buildings continue to have a spiritually calming role. They stand apart from the everyday world as places to use on dozy afternoons and special occasions.
As to whether a 'custom built log cabin' can fulfill the ancient role - we await the evidence. Manufacturers have however developed a range of high quality, even luxury, garden houses which are less expensive than buying the materials and doing the construction yourself. They need good insulation and good draught proofing. Well-designed timber buildings have these virtues.