So essentially a player is owned by a team until he’s 27 unless they trade his rights?
No, not necessarily, other posters listed out the various scenarios. Maybe to make it easy, let's just take this year's draft with a hypothetical scenario.
Team X wins the lottery but for whatever reason presumed #1 pick Shane Wright doesn't want to sign. He could hypothetically refuse to sign and re-enter the 2024 Draft at which point he'd be taken by Team Y who would own his rights for a couple years. Wright could refuse to sign, play an overage year in the OHL and beer league for another season, and then he'd be UFA in the summer of 2026 when he'd be required to sign a two year ELC. But that's pretty unrealistic that he'd give up four years of pro hockey and delay his ELC for that long.
He could refuse to sign and force a trade before he re-enters the 2024 Draft. Team X would only get the 33rd overall pick in 2024 as compensation, so it would be more beneficial to start a bidding war and trade Wright while they still own his rights.
Bryan Berard explains why he asked the Senators to trade him - Sportsnet.ca
It wasn't the same scenario, but Ottawa drafted Bryan Berard with the top pick in 1995 and there were complications with signing him (Berard wanted to play immediately, Ottawa wanted him to go back to juniors / didn't have the money to pay him) which led to some bad blood. Berard requested a trade and Ottawa dealt Berard in January 1996.
Carolina drafted Jack Johnson with the 3rd overall pick in 2005 back when people thought he was a mix of Scott Stevens and Scott Niedermayer. Carolina offered to sign him to an ELC during the spring of 2006 but Johnson was having a grand time in college and wanted to go back for his sophomore year. Apparently there was some animosity with the Johnson camp and Carolina regarding the negotiation which led to Carolina trading Johnson to Los Angeles on the eve of the eve of the 2006-07 season. Vaguely recall some threats that Johnson would play out his four years at Michigan and go UFA rather than sign with Carolina, but he signed with Los Angeles after his sophomore year.
Another posted listed out the Blake Wheeler scenario. Wheeler was an outlier in that he didn't start college hockey until his D+2 season and ended up spurning Phoenix who had drafted him 5th overall in 2004. Wheeler became UFA in the spring of 2008. But I think most considered Wheeler a raw project when he was drafted, so this is definitely a different scenario than a "generational" player who could step into the NHL at 18.
It's one thing for a raw prospect who was expected to need 2-3 years of development before turning pro to consider not signing. But it's a different question for a generational guy to sit out. As many others said, Eric Lindros is your closest example and he only sat out a year before Quebec decided to auction him off.