The 26-year-old has signed for the newly-promoted Mariners on a permanent deal, less than a year after moving to Wetherby Road from Maidenhead United on a two-year contract.
Despite netting six goals in 14 appearances for the Sulphurites during the first half of 2021/22, Orsi was loaned out to National League Boreham Wood until the end of last season.
He was then placed on the transfer-list upon his return to North Yorkshire, with Weaver stating that both he and the player were in agreement that a move elsewhere was for the best.
"He’s had a year of not playing too many matches with both us and Boreham Wood, so he’s in agreement with me. He desperately wants to start games," Harrogate's manager explained.
With Orsi's move to Grimsby confirmed on Friday afternoon, Weaver told the Harrogate Advertiser: "I think we've made it clear we back Luke Armstrong as that figure-head, he's physical and a goal-scorer.
"We’ve also got Mullers [Jack Muldoon] who is quick and his track-record is top-class.
"I've backed those two ahead of Danilo and we don’t hoard players."
On allowing Orsi to leave on a free transfer, Weaver added: "We let Danilo go for free as we didn’t want to delay him playing games.
"There aren't many fees going around at the moment as there are so many players around.
"His wage goes back into the budget and we let the lad go to a club that may feel he’ll get more game-time.
"Stick a fee in the way and it prevents him moving and we have a disenchanted player in the squad.
"Danilo is a great lad though, and we really mean it when we say that we want him to go and do well."
Orsi leaves Harrogate boasting a good goal-scoring record, having found the back of the net on six times in just six starts.
He opened his account for Harrogate with an EFL Trophy hat-trick in a 3-1 success over Mansfield Town before going on to register against Newcastle United Under-21s in the same competition, then hitting a late winner during November's 2-1 FA Cup victory at home to Wrexham.
His solitary League Two strike for the club came from the penalty-spot as the Sulphurites thrashed whipping boys Scunthorpe United 6-1 on October 9, but he did only start three league matches.
His spell in the National League with the Wood was however less impressive and ended with him having contributed just two goals in 24 matches.
Defenders Nathan Sheron and Leon Legge and striker Aaron Martin have also been transfer-listed by Town, but all three currently remain at the club.
"Shez is completing a move in a matter of days," Weaver added.
"There haven't been any offers for Leggey yet, though a few clubs have called him."