I don't live in Daegu, but I've been there many times. There's a variety of climbing in the city, from drive up and belay crags to hike in and trad/aid. . . (a search will lead you towards much, and I'd be up for exploring in the Spring). . . As Korea's 3rd city, and most central, you're not far from anything, on the KTX, train lines, airport, several bus terminals, a subway. . . lots of gyms, walls, stores, vegetarian restaurants. . . and hospitals (which unfortunately I've had a chance to sample as a patient and friend of patient!!)
that said, it has neither the culture of Seoul or the coast of Busan, but if the job gives you more time for family and climbing and better benefits. . . there are/have been many KOTRers living there (though many have since left) and planning a get together has brought out a dozen or so more on several occasions. There's also plenty of places within 2 hours.
If you do go. . . I'd really love a partner with extra days off during the spring. . . for trad, multi-pitch in the region . . . and I hear you might be good for that. . . in exchange, I will BABYSIT gladly during the rainy spring times so you and your wife can have some climbing (or otherwise) dates!
If you don't go. . . I also need someone for Seoraksan