Is there any particular reason why the devs never bothered with a means of bringing bazaar handling over to the server, rather than players keeping accounts logged in 24/7 which just puts more stress on the server?

It can't be that hard for them to have an NPC (or MH moogle) you can initiate before you log-off that keeps track of your currently bazaar'd items, and lists them on a dedicated section of the AH for bazaar'd items. The current means is just so archaic, especially with the client engine limits.