Five-time major victor Phil Mickelson, a three-time victor and fan favorite in Phoenix, shot 70 to enter the weekend at four-under par.
Matsuyama had a bogey-free round in the morning.
The world No. 5 carded four birdies at the Tournament Players Club before his only bogey of the day on the 18th saw him settle for a four-way tie for third, behind Brendan Steele of the United States and An Byeong Hun of South Korea.
There was no such concerns for Kuchar, though, whose five birdies and an eagle added up to his best-ever score at the TPC Scottsdale layout. His five birdies and an eagle are good enough to give him a one stroke lead heading into Friday.
"I look forward to playing those holes, 16, 17, 18 - great finishing holes", the South Korean player said.
"I knew I needed a break, and my break happened to be early on this year".
"I was able to be home, I was able to be dad".
Thousands came out for opening night at the Bird's Nest. Attendees came out for the music and to have a good time
"Hopefully through the week here I can get up to that same level and compete for the win again".
Asked how close he was to the scintillating form he produced in October to win the WGC-HSBC Champions by a record seven shots, he replied: "I'm not hitting it and putting as well as I did in Shanghai". My boys are 9 and 7 now, getting into golf. "Last week (at the Farmers Insurance Open) I was a bit shaky but Sunday I played really well, especially on the back nine".
Laird, who has three PGA Tour victories, shot a six-under par 65.
"I played so much this past summer. I still feel like today was very steady golf".
Matsuyama, who began last year's final round trailing by three strokes at 10-under, birdied the 72nd hole to post a 4-under 67 and force sudden death with Rickie Fowler (who is also in this year's field).
Kuchar is making his first start since the unofficial Hero World Challenge in early December previous year, but showed no signs of rust.