I think they can still send them down, but that player won't be eligible for AHL playoffs, no?
Nope. There are restrictions on Loaning players after the TDL.
CBA 13.12(j) A Player may be Loaned to a club of any league affiliated with the League at any time up to 3:00 p.m. New York time of the fortieth (40th) day immediately preceding the final day of the Regular Season (the "Trade Deadline").
Following this period, no Player may be Loaned, except that:
(i) a Player who has been Recalled during such restricted period under Article 13.12(l), 13.12(m), or 13.12(n) may be Loaned back to the member club of the affiliated league from which they were Recalled;
(ii) a Player who is on the Bona-Fide Long-Term Injury/Illness Exception, or who has been unfit for 24 calendar days and ten (10) NHL Regular Season games, may be Loaned on a Bona-Fide Long-Term Injury/Illness Exception Conditioning Loan (with the Player's consent) pursuant to Section 13.9 above for a period not to exceed the longer of six (6) days and three (3) games (which Loan is not subject to any extension), provided, however, if the Player is not on the Bona-Fide Long Term Injury/Illness Exception, he cannot be Loaned while he is on Injured Reserve; and
(iii) a Player may be Loaned (with the Player's consent) provided such Player (a) was on Recall and was determined to be disabled as of the Trade Deadline, (b) is determined to be fit to play prior to such Loan, and (c) was on the Active Roster for less than twenty-five percent (25%) of the period from the beginning of the Regular Season up to and including the date on which he was determined to be disabled.
Which breaks down to:
(i) Players recalled after the TDL can be returned back to the AHL.
(ii) Players on LTIR can go on a conditioning stint to the AHL.
(iii) Players who were disabled at the TDL can be loaned if they were on the Active Roster less then 25% of the season prior to the disability. Which essentially means players who were probably only on the team at the TDL by virtue of being injured.