Rules-as-written, the efreet cannot use wishes in this room, but this was not the rule or the intent of Tomb of Horrors releases for earlier editions of D&D.
AD&D
In the original AD&D Tomb of Horrors, the room does not explicitly prevent the efreet from using wishes. As in the 5th edition version, no spells or magic items work, except those that detect auras such as magic or evil, but this somehow does not prevent the illusion on 10,000 copper pieces in one of the chests, which have been enspelled to look like platinum.
In the AD&D edition, however, the efreet does not explicitly grant wishes, instead performing three "services", which may be implied to be wishes, as per the Monster Manual entry:
An efreeti can be forced to serve for a maximum of 1,001 days or by causing it to fulfil three wishes. They are not willing servants, and they will seek to pervert the intent of their masters by adhering to the letter of commands.
The word "wish" is not in italics, unlike the efreet's other spells, meaning that its ability to grant wishes in this edition was not considered a casting of the spell wish. Since the anti-magic only affects spells and items, the efreet in Gygax's Tomb of Horrors could originally grant wishes in the room.
D&D 3rd edition
In the earlier D&D 3rd edition version of Tomb of Horrors, the Efreeti has a unique special ability to allow its own special abilities, including wish. This explicitly resolves the conflict between the anti-magic effect, which in D&D is a variant antimagic field spell effect, which in that edition blocks spell-like abilities including the efreet's wish.
The illusion on the contents of the chests in the 3e Tomb of Horrors are also explicitly stated to be immune to the effects of the antimagic.
In short, the efreet in Bruce Cordell's third edition Tomb of Horrors could also grant wishes in the room.
D&D 5th edition
In this edition of Tomb of Horrors, no spells function within this room. The description of genie wishes in the Monster Manual specifies that genies specifically grant wishes by casting the wish spell. I strongly suspect that Jeremy Crawford, with his literal rules interpretations, would insist that this is the case.
The efreet is not bound to the room, and clever players might notice the antimagic and invite him outside after the first wish fails. However, the Monster Manual also notes that certain genies will pervert the intent of their wishes by adhering to the letter of the words, and the lawful evil efreet are most likely to do this, especially given earlier lore which says efreet are very specifically the ones who do this.
We have, in essence, a conflict between two statements in the same section: no spells work within the room, but the efreet does grant three wishes, and in earlier versions these were not considered mutually exclusive. Like all ambiguities in D&D, that's up to the DM to adjudicate, meaning of course that your approach was valid and within the rules. In fact, within the lore, the efreet may intentionally remain within the antimagic room just to cheat the player characters out of their wishes.
That door in the Tomb of Horrors has often traditionally been one-way as the question speculates it likely should be, although exact details of that door vary by edition.
Tomb of Horrors (1978) says this about that door:
Adamantite Door: Although it is marked secret, it is very evident; the marking is simply to make certain that its actual nature is known. It has permanent anti-magics on it, and there is no magical or physical way of forcing entry. There are 3 slots in the door at about waist height. If 3 sword blades are shoved simultaneously into the slots, the 1' thick panel will swing open. This is a one way door which cannot be prevented from closing in 5 rounds! (7)
(Changes from all caps to boldface mine.)
Return to the Tomb of Horrors (1998) includes this:
During Return to the Tomb of Horrors the PCs venture into the adnd Tomb, a facsimile of which is included with the Return box set. The adventure expects the adnd-2e DM to update the original Tomb to the new edition; Return to the Tomb of Horrors—the adventure proper rather than its ancillary books—has more details on page 55, none changing this door.
Tomb of Horrors was not to my knowledge otherwise updated for this edition.
Tomb of Horrors (2005) says that at the end of that corridor the adventurers see this:
A door forged of gleaming metallic alloy with massive reinforced hinges bars passage beyond this point. Three vertical slots mar the door’s surface at waist height. Each slot is about 1 inch wide and 3 inches long.
That's followed by information for the Dungeon Master:
The 1-foot-thick steel door (it’s too expensive for the demons to keep replacing adamantine doors) is suffused with a globe of invulnerability effect. (However, only the door, its hinges, and the stone around the hinges are so affected, allowing it to shed any spell of 4th or lesser level.) The effect cannot be brought down by a targeted dispel magic, but it can be suppressed for 1d4 rounds if the dispel check beats a DC of 22. If the door is removed from the stone that moors it through some determined engineering, it loses all magical abilities. The door is locked (DC 45 Open Lock check), but opens of its own accord if three sword blades are simultaneously shoved into the slots.
(Links mine.) Then follows another note for the Dungeon Master:
Once open, the door automatically swings closed 5 rounds later. In addition to the invulnerability effect, the door enjoys a magical hardness (which can be suppressed for 1d4 rounds as the invulnerability effect) that allows it to mash any metal less deformable than adamantine that someone may put in place to hold it open. A character trying to hold it open needs to make a DC 30 Strength check each round to hold it open. From inside (room 25), no slots or other obvious methods can get the door open again. (25)
I've not read the Greyhawk Classics novelization Tomb of Horrors (2002) that was released during the Dungeons and Dragons, 3rd Edition era, so I don't know if this door is described differently therein.
Tomb of Horrors (2010) says this about that door:
Nothing! Building as it does on the 2nd Edition adventure Return to the Tomb of Horrors, the original tomb's layout is duplicated, but time has seen its guardians replaced and many of its traps defeated. Although the included map does designate the door as secret, sadly no special features of it are described. (See Tomb of Horrors on pages 100 and 154–5 for the scant details.)
Wikipedia's Tomb of Horrors entry says that a more direct port of the original adventure exists for 4th Edition, calling it an "update of the original module for 4th Edition rules, [that was] written by Scott Fitzgerald Gray and [that was] released to members of the RPGA as part of the DM Rewards program." I don't have access to this adventure.
As a more-or-less direct port of the original, this reader agrees that the text is likely in error and that the door should prevent those who pass through it from returning whence they came.
Best Answer
Each of the seven doors is opened by one method, not all seven.
The text in the 5th edition is ambigious, and may be interpreted that each door requires a seven-step process to open. However, if you look at the map, each door is lettered, making it clearer that, for example, step "A" is the method for opening the door "A", step "B" the second door, and so on.
This is made much clearer in the original AD&D module:
The D&D 3.5 version concurs:
Further, it makes this note, defending the use of trial-and-error rather than skill checks: