Your description isn't exactly true, as you are omitting a key provision in the waivers process.
While it's true that if a team puts a player on waivers, then re-acquires him via waivers during the same season, that player can be sent down without having to clear waivers again. However, it is only true if the original team is the ONLY team that makes a waiver claim on that player:
Quoting: NHL CBA Section 13.22
When a Club claims a Player on Regular or Unconditional Waivers, and, subsequently, in the same season it requests Waivers on the same Player and the original owning Club is the successful and only Club making a Waiver claim, then the original owning Club shall be entitled to Loan such Player to a club in another league within thirty days without further Waivers being asked; provided that such Player has not participated in ten or more NHL Games (cumulative) and remained on an NHL roster more than thirty days (cumulative) following such successful claim.
So let's say Ottawa took Sprong off waivers last week, decided they didn't want him, and put him back on waivers. The Ducks, having a relatively high waiver priority spot, put in a claim, but so does one or more teams with a lower priority. The Ducks then re-acquire Sprong, but wouldn't be allowed to demote him to the AHL without having him clear waivers again.
That provision exists so that two teams don't keep fighting over the same player during the course of a season and gives demotion rights to the original owning team. However, if there's another team in the league that has interest in playing that player in the NHL, that team would still get a shot at claiming him on waivers prior to the Ducks being able to demote him to the AHL.