Excellent question: The answer is at the end of this post. The first part may help to introduce the topic.
The mishna statued in several points that hole is as a doorway, more than 10 amot,the wall is kicked and annulled on this area.
See Mishna daf 15b
פירצה שהיא כעשר אמות מותרת מפני שהיא כפתח יתר מכאן אסור:
any gap which [in its width does not exceed] ten cubits is permitted, because it is like a doorway. if it exceeds this [measurement] it is forbidden.
See Mishna daf 2a, not about a hole in wall, but about an alley ([a cross-beam spanning] the entrance [to a blind alley] (in Soncino translation)):
והרחב מעשר אמות ימעט ואם יש לו צורת הפתח אע''פ שהוא רחב מעשר אמות אין צריך למעט:
and [any entrance] that is wider than ten cubits should be reduced [in width]; but if it has the shape of a doorway there is no need to reduce it even though it is wider than ten cubits.
Gemara daf 11a:
דתניא: מבוי שהוא גבוה מעשרים אמה - ימעט, והרחב מעשר - ימעט, ואם יש לו צורת הפתח - אינו צריך למעט, ... אמר ליה: אתנייה (רב) צריך למעט.
Come and hear what was taught: [A cross-beam spanning the] entrance [to a build alley] that is wider than ten cubits should be reduced [in width], but if it had the shape of a doorway, there is no need to reduce [the height of the beam] and if it 'has a cornice there is no need to reduce'… Teach him, [Rab] said to him, It is necessary to reduce it.
Tosfot daf 14a wrote:
יש להוכיח דאין הלכה כרב דקאמר לעיל (ד' יא.) אתנייה צריך למעט דהמשנה לא נשנית כמו שהגיה וכאן נשנית כמו שהגיה וכן פסקו הגאונים
Rabenu Tam and Geonim learn that the halacha does not follow Rav's opinion. A gap greater than 10 cubits is allowed with a cornice. Rabenu Tam explain that the Mishna has been written following the conclusion of the Gemara.
Gemara daf 11a
אמר רב יוסף מדברי רבינו נלמד חצר שרובה פתחים וחלונות אינה ניתרת בצורת הפתח מ''ט הואיל ויותר מעשר אוסר במבוי ופרוץ מרובה על העומד אוסר בחצר מה יותר מעשר האוסר במבוי אינו ניתר בצורת הפתח אף פרוץ מרובה על העומד האוסר בחצר אינו ניתר בצורת הפתח
Let us explain thit statement of Gemara according to Rashi.
- Rav ruled that an entrance wider than ten cubits causes the prohibition of an alley.
- Paruts merube Al haomed is prohibited in
- a breach [in a wall] that is larger than its standing [portions] causes the prohibition of a courtyard
- the two may be compared
- an opening that is] wider than ten cubits, which causes the prohibition of an alley cannot be ritually rectified by means of the shape of a doorway
- so also a [wall] the breach in which is larger than its standing [portions], which causes the prohibition of a courtyard, cannot be ritually rectified by means of the shape of a doorway.
- The rule on the first paragraph is not accepted as we saw above in name of Rabenu Tam
But Rambam Shabbat 16, 16 wrote
Any partition whose open portion exceeds its closed portion is not considered to be a partition If, however, the open portion is equivalent to its closed portion, it is permitted [to carry within the enclosure], provided none of the open portions is larger than ten cubits wide. [The rational for this leniency is that an open space] ten cubits [or less] is regarded as an entrance.
If, however, this open space [is enclosed by] **the frame of an entrance**, even if it is wider than ten cubits the partition is not considered to be nullified, provided the open space does not exceed the closed portion.
- At this point we do not discuss the rules of paruts merube Al Haomed with Tsurot Hapetach
- Shulchan Aruch rules as Rabenu Tam. More than 10 too is allowed with a doorway and paruts merube
See a consequence of this about paruts merube: later in Gemara 11a:
ואף ר' יוחנן סבר לה להא דרב דאמר רבין בר רב אדא א''ר יצחק מעשה באדם אחד מבקעת בית חורתן שנעץ ד' קונדיסין בארבע פינות השדה ומתח זמורה עליהם ובא מעשה לפני חכמים והתירו לו לענין כלאים ואמר ר''ל כדרך שהתירו לו לענין כלאים כך התירו לו לענין שבת ר' יוחנן אמר לכלאים התירו לו לענין שבת לא התירו לו
Rabenu Tam rules as Resh Lakish. And S.A. too, and even with doorway large than 10 cubits. But it is not allowed in Vallee.
For Rabin son of Rav Adda stated in the name of Rav Isaac: It once happened that a man of the valley of Beth Hiwartan drove four poles in the four corners of his fiel and stretched across [each two of] them a rod, and when the case was submitted to the Sages they allowed him [its use] in respect of kil'ayim.
And [in connection with this statement] Resh Lakish remarked: As they allowed him [its use] In respect of kil'ayim so have they allowed it to him in respect of the Sabbath, but Rabbi Johanan said: Only in respect of kil'ayim did they allow him [its use]; they did not allow it in respect of the Sabbath.
- See Shulchan Aruch Orach Chayim 362,10 and MISHNA BERURA
But the maximum leniency, paruts merube and tsurat hapetach for more than 10 cubits doesn't help for the OP which regards doors without tsurat Hapetach.
Gates without Tsurat Hapetach:
If there isn't tsurat hapetach, the only relevant point is the nature of the pirtsa, and since the door moves and is closed by night.You can close the pqirtsa when you want, so this is not a pirtsa. Similarly The Gemara daf 6b says that because of its gates which are closed by night, Yerushalayim is not a public place, why? Because it's not paruts larabim this has nothing to do with the tsurat hapetach (which is not relevant there). This was my svara.
Thanks to DoubleAA comment, I read in Chazon Ish that a door without Tsurat Hapetach is nothing neither for hilchot mezuza, neither for hilchot shabbat for hilchot Shabbat, see here SIMAN 78 the original text.