The plural expression is common in Hazalic literature that refers to divine involvement with man, thus for example, Shabbat (104a) which famously states that man is allowed to sin, but helped to purify himself:
בא ליטמא - פותחין לו, בא ליטהר - מסייעים אותו
One who comes to become impure, they allow him, one who comes to become pure, they help him.
For many additional examples, see the bottom of this post.
Regarding Berakhot 10b, Meiri (there) (who perhaps had a variant text) actually writes exactly what you propose; that feeling bad for a sin atones for that sin:
כל העושה דבר ומתבייש בו מוחלים לו על אותו דבר
Anyone who does something and is embarrassed about it, is forgiven for that thing.
The Havruta there (note 40) cites Ritva (If anyone has access to it, please edit it in) who similarly writes that every sin that a person does is forgiven if he feels about it.
Importantly, a very very similar passage in Hagiga (5a) states precisely that; that one who regrets a sin is forgiven for it:
אמר רבי חנינא בר פפא: כל העושה דבר ומתחרט בו - מוחלין לו מיד
However, the plot thickens for R. Yosef Gikatilla quotes the passage in Hagiga in Klalei HaMitsvot (חרט) as stating that all sins are forgiven! This same version of the Gemara Hagiga appears elsewhere in his book (כפרה). Furthermore, while many Rishonim and some manuscripts (e.g. Munich 95, Oxford Opp. Add., London - BL Harl. 5508 (400), and Goettingen 3) support our reading of Hagiga that it is that particular sin that is forgiven, R. Gikatilla's reading is also supported by some manuscripts (Munich 6, Spanish Print (c. 1480), and Vatican 134 which states מוחלין לו עוונותיו)
Although it is possible that the Meiri and Ritva had a variant text, in accordance with the passage in Hagiga, I have found no variant manuscripts of the passage in Berakhot, and Munich 95, Oxford Opp., and Paris 671 all state that all sins will be forgiven. It is therefore also possible that Ritva and / or Meiri had the same text but explained that every sin can be forgiven by feeling bad about it, not that feeling bad for one will atone for all.
Alternatively, the Havruta suggests that the embarrassment leads to atonement for other sins, since it is a form of suffering. They cite R. Hayyim Shmulevitz (31:5, 32:11) who writes that the purpose of suffering for sins is to take Man off his high horse, once one feels embarrassed for his sin, this is accomplished, and further suffering is unnecessary:
שכל מטרת היסורים הם לבייש את האדם, כי הם מגלים על פחיתות כוחו, וע"י העונש נוכח בטעותו, והמתבייש בעבירה הגיע מעצמו להכרה זו שהיא מטרת היסורים, ואינו צריך לעונש נוסף
Additionally, for context, it should be noted that elsewhere Hazal state that all of ones sins to be forgiven. For example, in reward for "letting it go after one bothers you", Rosh HaShana (17a) and Yoma (23a) state: מעבירין לו על כל פשעיו or according to the text in Megillah (28a):
מעבירין ממנו כל פשעיו. (As a side note, note the plural expression). Similarly, note Pirkei D'Rabbi Eliezer (ed. Higger ch. 19) that if one observes Shabbat all of his sins will be forgiven. Note also Alpha Beta of Ben Sira in Otsar HaMidrashim (p. 38) that of one gives charity all of his sins will be forgiven. A similar statement is found in Midrash Iyyov (in Batei Midrashot Vol. II: 45). Other Hazalic sources also throw around expressions about all of one's sins being forgiven, see e.g. Vayikra Rabba (ed. Margolis ch. 19). While these might all be literal, perhaps it is a common phrase used as an exaggeration.
Additional examples of the plural usage in Hazal to refer to divine involvement with man are the following:
The bond between the yavam and the yevama is automatically (divinely) created. The rabbis refer to this in Nedarim (10:6) as a wife which he got from heaven, again using the plural:
אשה שהקנו לו מן השמים
This is expression is also used in Ketubot (82b), Yevamot (39a), and Nedarim (74a).
Similarly, note Berakhot (55a) that there are three things things whose performance leads to divinely long life, again using the plural:
ואמר רב יהודה שלשה דברים [המאריך בהן] מאריכין ימיו ושנותיו של אדם
Similarly, Berakhot (57a) states:
הרואה הספד בחלום - מן השמים חסו עליו ופדאוהו
Similarly, note Hagiga (5b) that states that "they" recount to a person his speech to his wife:
אמר רב: אפילו שיחה יתירה שבין איש לאשתו מגידים לו לאדם בשעת מיתה.
Similarly, Shabbat (127b) states that if one judges other favourably, "they" judge him favourably. From context is clear that this refers to God, as noted by Sfat Emet there:
הדן חבירו לכף זכות דנין אותו לזכות
Similarly, Sotah (1:7) states that "in the measure that a person acts, so too "they" mete out to him:
במדה שאדם מודד בה מודדין לו
This statement is found in Tosefta Sotah (1:7, 3:1, 4:1) and in the Talmud Megillah (12b), and in Sotah (8b), and in the Yerushalmi Sotah (1:7).
Similarly, Tosefta Sh'vuot (3:1) states that "they are not obligated to pay according to the law, but 'they' do not forgive him in heaven, until he pays."
אין חייבין לשלם מן הדין ואין מן השמים מוחלין להן עד שישלמו
The same expression is found later in the Tosefta there (3:2, 3:3, 3:4).
Similar usages are present in Berakhot (55b), Shabbat (129a), Pesahim (66b), Taanit (14b), Moed Kattan (15a), Yevamot (42a), Sotah (2a), Bava Kamma (55a), Bava Metsia (59b), Hulin (92a), Yerushalmi Yevamot (16:3), Sifrei (B'haalot'kha 78), Sifrei Zuta (11:16), Sifrei (Ekev 40), Sifrei (R'eh 96), Leviticus Rabba (ed. Margolis): (Tsav 7), (Aharei Mot 22:6), (Emor 26), (Emor 30), (B'har 34), Ecclesiastes Rabba (1), P'sikta D'Rav Kahanna (5:3 in ed. Mandlebaum), Mishnat R. Eliezer (15: pg. 286), and Kitv'ei Midrashim Genizah (Mann) (tet: Kitsur Shel Derashot Yelamdenu pg. 76).