Am I mistaken that players who were passed over in the recent draft must go through the draft next season, and can't be signed?
Depends on where they played last season. If they played (exclusively) in North America, then they could sign ELCs and forgo the draft. It's not unusual to see a handful of undrafted guys earn ELCs. Philippe Myers got one with Philadelphia last year and Columbus did it with Kole Sherwood. Martin Jones is probably the most notable guy who went undrafted (2008) but earned an ELC as a prospect camp invite and didn't have to go back into the draft.
Usually a bit easier to do it with the CHL guys since they can sign contracts and go back to their junior clubs. NCAA players would be giving up their eligibility to sign a deal.
If a guy played in Europe last season, he's not allowed to be signed. I know that the Kings had Radko Gudas as an undrafted invitee for their 2009 camp and they really liked him, but he was off limits. The Flyers goofed and tried to sign Tomas Hyka, but the league rejected the contract. Ditto the Oilers with Vladimir Tkachev a couple years back (Tkachev had played a couple of games in Russia before jumping to the QMJHL).
From a quick scan of our invite list, I think only McInnis/Berzolla have remaining draft eligibility anyways.