Spieth was in the early feature group with Koepka and the year's other major victor, Spain's Sergio Garcia, the Masters champion.
Still, Spieth didn't shoot himself out of the tournament, making birdie on the par-5 seventh and another on the eighth after stuffing a wedge to two feet.
The 21-year-old Olesen set the pace with a four-under 67 while Spieth, the tour's best putter for much of short his career until this year, said he could not have putted any worse en route to a one-over 72 in calm morning conditions.
"Historically I'm pretty solid with the lead so that was kind of the goal", Spieth said. Some were good putts and some weren't. "I mean, that's never fun to walk up and see somebody, you just drilled them".
Northern Ireland's Rory McIlroy was one of the late starters at Quail Hollow.
"I really just haven't got the results I've been hoping for and I'd put it more down to poor driving than anything by my standards".
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"I don't think I missed any short putts today".
Spieth had insisted on Wednesday that he had no "burning desire" to surpass Woods as the youngest grand slam victor and was "freerolling" this week, but added: "I don't think I was as freerolling as I thought I would be, as you can tell by some frustration".
The world number one mixed three birdies with two bogeys on Thursday to card a one-under 70 at the year's final major.
"I feel like I played nicely", he said. It was a big advantage to be in the fairway, and Olesen had only a 7-iron into 18th green.
Ever since the creation of golf more than 500 years ago, the argument has raged on - are golfers actually considered athletes?
While wearing what appears to be a sportjacket from the Roy Williams Collection and a couple of gold chains, the Aviator-wearing Flair gets you hyped for some major championship golf.