The Tur (OC 46) had the same issue as you. He writes that the custom around him was to say Birkot haTorah and some Torah content before Pesukei deZimra but after reciting the "Leolam Yehei" and "Attah Hu" prayers which themselves contain Torah content including verses. Accordingly he changed his own practice to say Birkot haTorah immediately after Elohai Neshama at the very beginning of the day.
However, other Rishonim (such as Agur (92)) do not have a problem with this set up because the verses are being recited דרך תחנונים by way of supplication, not דרך לימוד by way of learning. Indeed, by the fact that the Tur thought to change it, we see longstanding custom was in accordance with this view. The Gra records in his commentary to the below-cited Shulchan Aruch that in all the very old Siddurim or records of the order of prayer he could find, he saw that all of them place Birkot haTorah later.
The Shuchan Aruch (OC 46:9) rules:
לא יקרא פסוקים קודם ברכת התורה, אעפ"י שהוא אומרם דרך תחנונים. ויש אומרים שאין לחוש, כיון שאינו אומרם אלא דרך תחנונים, ונכון לחוש לסברא ראשונה. הגה: אבל המנהג כסברה אחרונה
One should not read verses before reciting Birkot haTorah, even though they are being said by way of supplication. And some say one should not worry about this because they are only said by way of supplication. And it is proper to worry about the first opinion. Gloss: But the custom is like the latter opinion.
So it seems there is strong traditional basis to not worry about saying verses by way of supplication before reciting Birkot haTorah.
Now for the other position (which many Achronim view as a common/proper Chumrah (stringency) and not Ikar haDin (letter of the law)), I can imagine a number of possibilities as to how one might end up saying the blessings of Shema without reciting Pesukei deZimra:
You are late to minyan and skip all of Pesukei deZimra (ShA OC 52:1)
You are stuck and can't daven Shacharit immediately after Keriat Shema but can later (ShA OC 64:10)
You are exempt from Tefilla but not Keriat Shema (ShA OC 106:1)
You against your will were unable to recite Keriat Shema at night and are saying it in the morning before sunrise (ShA OC 58:3)
At the very least the rule is relevant in case someone has already said Birkot Keriat Shema, even if you want to say he shouldn't have done so.