In a statement, Mr Lynn's family expressed their distress over the case, while acknowledging the anguish of the campers' families.
"It has been distressing for our family and the families of the deceased and we are completely traumatised as well by these tragic events," they said.
"We need to deal with the legal proceedings as they arise and try to restore some balance in our lives.
"We also acknowledge the suffering of the Clay and Hill families at this difficult time."
The family said they understood the media were interested in the proceedings but asked for privacy.
The Jetstar pilot appeared calm when he fronted Sale Magistrates' Court via video link this morning.
He wore a mask and held his hands together, taking off his mask occasionally to take a sip of water.
His wife Melanie Lynn also tuned in remotely.
He did not apply for bail and will remain in custody until May 31, 2022.
Mr Lynn was taken away from court in a prison van, which arrived at the Melbourne remand centre this afternoon.
Ms Clay, 73, and Mr Hill, 74, disappeared while camping in the Wonnangatta Valley on March 20 last year.
Victoria Police Assistant Commissioner Bob Hill said yesterday police were yet to find the remains of the missing pair.
He said police had located a specific area of interest in the Greater Alpine region in the search for the remains, and that police were hopeful they would find them.
"Our efforts to locate the remains of Russell and Carol will continue," he said.
"While this arrest is a testament to months of dedication and tireless work from detectives ... it's far from over.
"Police are hopeful we will be able to locate the deceased and provide closure to the families."
Forensic specialists will commence their search among bushland over the coming days.
Assistant Commissioner Hill said investigators had established the pair were deceased by mid-2020.
Police also made it clear they were not looking for anyone else in relation to the investigation.
However, they are looking for a metal silver-blue trailer, which they say Mr Lynn was towing around the time the campers disappeared.
"We believe the trailer was sold on Gumtree," Assistant Commissioner Hill said.