Your impression of takfīr as a hugely risky process is indeed correct. The Prophet ﷺ warned his companions, and the rest of the Muslim nation, of the risk involved of publicly declaring one as a disbeliever is that one (or both) of them bears the title (either the one declaring or the one being declared against):
حَدَّثَنَا أَبُو بَكْرِ بْنُ أَبِي شَيْبَةَ، حَدَّثَنَا مُحَمَّدُ بْنُ بِشْرٍ، وَعَبْدُ اللَّهِ بْنُ نُمَيْرٍ، قَالاَ حَدَّثَنَا عُبَيْدُ اللَّهِ بْنُ عُمَرَ، عَنْ نَافِعٍ، عَنِ ابْنِ عُمَرَ، أَنَّ النَّبِيَّ صلى الله عليه وسلم قَالَ: إِذَا كَفَّرَ الرَّجُلُ أَخَاهُ فَقَدْ بَاءَ بِهَا أَحَدُهُمَا
It is reported on the authority of Ibn 'Umar that the Apostle (may peace and blessings be upon him) observed: "When a man calls his brother an unbeliever, it returns (at least) to one of them."
— Sahih Muslim, Book 1, Hadith 121
This also implies that doing so may be required at one stage or another. The hadith is a deterrent, not a prohibition. It needs to be done with caution and by those who are qualified; yet, the fact remains that there are reasons for takfīr to be done. This is what makes your impression of takfīr being pointless to be incorrect. Matters of marriage (or its continuation thereof), funeral and burial process, inheritance, among other liabilities, are entirely dependent on one's belief either as the single criterion or as one of the major criteria. It is not a matter to be taken lightly either not to exercise takfīr. However, as your reference said on the same page (pp.120): "this principle is not unrestricted."
The concept that you are asking about "Whoever does not make Takfīr to the mushrikīn, or doubts their kufr, or approves of their Math'hab, he has disbelieved" predates Muhammad ibn 'Abdul-Wahhāb (1115 A.H.-1206 A.H.). For instance, Qadi 'Ayyad (Maliki, 476 A.H.-544 A.H.), more than six centuries earlier, said in his book Ash-Shifa:
ولهذا نكفر من لم يكفر من دان بغير ملة المسلمين من الملل أو وقف فيهم أو شك أو صحح مذهبهم وإن أظهر مع ذلك الإسلام واعتقده واعتقد إبطال كل مذهب سواه
— NOTE. My own translation, so treat with care.
This is why we declare as a disbeliever whoever does not consider non-Muslims as disbelievers or declines to declare them as disbelievers or doubts they are disbelievers or justifies their belief; even if he publicly declares he is a Muslim while genuinely believing in Islam being the only true religion.
— Ash-Shifa bi Ta'rīf Huqūq Al-Mustafa, Vol. 3, pp. 286 (Arabic only)
There are numerous other scholars who have reiterrated the same concept, e.g., An-Nawawi (Shafi'i, 631 A.H.-676 A.H.) in his book Rawdat at-Tālibīn 10/70 (Arabic only) said that one "who does not call a non-Muslim a disbeliever, e.g., Christians, or doubts their disblief or approves of their belief, then he himself is a disbeliever, even if he says he is Muslim, and even if he believes that Islam is the correct belief," and Al-Bahūtī (Hanbali, 1000 A.H.-1051 A.H.) in his book Kashshāf al-Qinā' 6/170 (Arabic only) said the same thing effectively, and so on.
The origin of this "third nullification" comes from the governing principle that whoever denies what is known by necessity or by proof of part of the religion is a disbeliever as Ibn Hajar al-Haytami (909 A.H.-974 A.H.) said:
يعني ما قلنا إنه واجب أو حلال أو حرام معلوم من الدين بالضرورة من حيث أصل كل منها، وإن وقع خلاف في بعض تفاصيل صور من العملي، فمن أنكر واحدا منها بالكلية أو اعتقد وجوب ما ليس بواجب بالإجماع كصلاة سادسة، اعتقد أن وجوبها كوجوب الخمس فخرج نحو الوتر أو أنكر مشروعية السنن الراتبة أو صلاة العيدين أو أنكر بقية الصلاة، زاعما أنها لم ترد إلا مجملة كفر
— NOTE. My own translation, so treat with care.
What we said previously means that what is known to be mandatory or halāl (permissible) or harām (forbidden) by proof from the religion based on the principle(s) for each [classification], even if there is a difference in some of the details of its appication, then whoever denies one in its entirety or believes mandatory what is not by consensus — e.g., having a sixth prayer that he believes is as mandatory the five main prayers, as in the witr prayer), or denies al-sunan ar-rātiba (the proven and confirmed nafl prayers) or the prayers of the two Eids — claiming that they are not stipulated, then he is a disbeliever.
— Al-Fatāwa al-Hadithiyya, pp. 142 (Arabic only)
Finally, takfīr is indeed a huge task that can only be undertaken by a knowledgeable body of a certain level of scholars (mujtahid levels); it is not a matter that followers or commoners (see the classification of seekers of knowledge for more information) should undertake. This was clearly mentioned by Ibn Taymiyyah (661 A.H.-728 A.H.), and of course Muhammad ibn 'Abdul-Wahhāb, in his book Majmū' al Fatāwa:
وغالب مذاهب الأئمة فيها تفصيل وحقيقة الأمر في ذلك: أن القول قد يكون كفرا فيطلق القول بتكفير صاحبه ويقال من قال كذا فهو كافر لكن الشخص المعين الذي قاله لا يحكم بكفره حتى تقوم عليه الحجة التي يكفر تاركها
— NOTE. My own translation, so treat with care.
And most of the madhhabs of the Imams are to be restricted and the truth is: A saying [principle] may be disbelief, therefore its owner is declared as a disbeliever. It is said "whoever says so is a disbeliever," but naming a person as a disbeliever is not to be deemed valid until proof is established against him, then he would be declared a disbeliever by ignoring it [the proof].
— Majmū' al Fatāwa, Vol. 24, pp. 345
You may refer to this Islam Q&A articles on guidelines on ruling someone to be a disbeliever and evidence that the one who is ignorant is excused with regard to issues of shirk and kufr for more information.
Note. Apart from Wikipedia and the media, I have not come across any credible Sunni source that refers to Muhammad ibn 'Abdul-Wahhāb as the founder of Wahhabism. I am not sure where else you may have come across this term, so I assumed in this answer the position of the definition in Wikipedia, which the media has often taken to declare Ibn Taymiyyah, who ironically died a few centuries before Ibn 'Abdul-Wahhāb, as one of his followers.
