According to the Montreal Gazette "Players can move freely between the taxi squad and the NHL roster, but call-ups must be made prior to 5 p.m. on game day." I don't believe you need waivers or be injured to move back and forth after the season starts.
If you think about it, why have a taxi squad if the playerrs can't move up and down freely'
Yes, Taxi Squad players can move freely IF there is an injury/covid issue. But to move other players to the squad (ie. Nielsen) they need to pass through waivers. You can't just pass anyone back and forth at any time.
At the moment we have 6 designated taxi squad players, Chowolski, Rasmussen, Lindstrom, Smith, Boyle, and Pickard. The first four are waiver except, Pickard and Boyle are not so they had to pass through waivers to be put on the taxi squad on January 11th. From here forward, if someone on our roster is unable to play, any of these guys can play for them. If the Wings wanted to put say Neilsen on the squad and permanently bring up one of the kids, they would have to pass Nielsen through waivers first, then add (say Pickard) to the main roster. Then if they wanted to bring Nielsen back to the main squad, they would have to waive Pickard (or someone else) to do it.