With the Regnum Carya Golf Resort & Spa hosting the tournament for the first time, Otaegui's 63 had earlier eclipsed the course record established by first-round leader George Coetzee, who could only add a 73 to his opening 64.
Willett, who needs to finish outright fifth or better to reclaim top spot in the Race to Dubai from Henrik Stenson, fared even worse with a 74 which left him 16 shots off the lead and in a tie for 58th in the 78-man field.
Olesen was in contention for a Ryder Cup place after winning the Alfred Dunhill Links Championship last year and making a strong start to this season, but had missed the cut in seven of his last 10 events before this week.
The 26-year-old was invited to Hazeltine anyway by fellow Dane Thomas Bjorn, one of European captain Darren Clarke's vice-captains, and now has extra motivation to qualify for the team which will try to regain the trophy in Paris in 2018.
"It was a great experience," Olesen said. "In the beginning it was a little bit difficult to be there when I had a pretty good chance in the summer to make the team, but I felt like I learned a lot being around the captain and all the vice-captains.
"One of my goals is to make the team and if I make it I will be more prepared and know a little bit of what I'm going into.
"It's so different from any other tournament and I realised that when I got there. It's so much bigger than you think and I was standing on that first tee on Friday to see the atmosphere there and get a bit of that feeling.
"The crowd were really rough. Some of the stuff [shouted by fans] was very personal and I thought it was hard for the players. I'm happy I saw that because that's how it going to be in America in the future as well."
Germany's Martin Kaymer enjoyed a similar experience to Olesen at Valhalla in 2008 and famously went on to secure the point which retained the trophy at Medinah in 2012.
"I hope to be part of the team," Olesen added. "Obviously holing the winning putt would be great, but as long as we win that's the most important thing. I think I can be there. If I keep playing well there is a good chance."
Willett's Ryder Cup debut ended in disappointment after three defeats from three matches amid the furore caused by an article written by his brother Pete which criticised the behaviour of American fans.
And the 29-year-old is also in danger of missing out on the Race to Dubai title for a second year in succession after failing to card a single birdie in his second round in Antalya.
Masters champion Willett had led the money list since claiming his first major title at Augusta in April, but was overtaken by Stenson on Sunday after the Open champion was joint second in the WGC-HSBC Champions and Willett finished 75th in the 78-man field.
That turned Stenson's deficit of more than 400,000 points into a lead of 261,387 heading into the three events of the Final Series, with both men set to play in South Africa next week and the season finale in Dubai.
Willett had been encouraged by an opening 69 and hit every fairway in regulation on the front nine on Friday, but failed to get up and down from just left of the sixth green and dropped further shots on the 16th and 18th.
Willett refused to speak to reporters at the end of his second round, but later tweeted: "Well this game is really kicking me in the nuts at the minute!! If it's not one thing it's another.. Haha.. #grinditout #TurkishAirlinesOpen"