>>16297tl;dr the existence of the game is a huge plothole, and there is no way they could take it that it'll make sense while respecting the lore
Witchers in-universe are genetically mutated abominations that need to undergo a very specific form of gene therapy as they're undergoing puberty - in turn changing a non-magic attuned human into somebody who can use basic cantrips, is attuned to vague magic current and able to digest neurotoxin-grade potions - and thus is able to fight magic-based monsters (caused by an event where dimensions overlapped, leaving humans and monsters stranded in a world they don't belong in). Most Witchers are failed creations in some way that simply "survived" and have incomplete powers - they're also sterile, emotionally numb, feel ill almost all the time and many have significant disabilities. Geralt is the one-in-a-million outlier success story who turned out strong + complete, and even he still has all of the downsides. Most children die during this process because of genetic mismatches and fucking up the difficult alchemy involved, and the Witcher 3 has an important main questline emphasizing that the School of the Wolf
(Geralt's school, and the most stable + safe + balanced of the versions of the process) is functionally extinct, after the last person who knew how to do the process dies, and the equipment to do it is destroyed. The takeaway is that the remaining 2-3 Witchers all think "good riddance", since the tradeoff is that monsters are going extinct anyways as humanity develops, and a witcher's work is becoming increasingly scarce and dangerous as the few remaining monsters are less the "shambling zombie nuisance" types and more "the pissed off serial killer demonbeast the size of a schoolbus that can blend into crowds with polymorphing" types.
There are however, other Schools of Witchers. Most of them are also extinct, for basically the same reason
(there's just not enough work left + witchers are socially untrusted outcasts that nobody likes). On top of the Wolves (Geralt's), the Bears + Gryphons + Manticores are all extinct. The only non-extinct schools are the Cats and the Vipers. The Viper school (focused on alchemy and poisons) is the entire basis of the Witcher 2, with the last Viper turning traitor and trying to appeal to an invading empire to let him refound his school as part-time Witchers, part-time supersoldier assassins (for the sake of having some stock of Witchers to deal with the remaining monsters - now with imperial funding to "finish the job", and a long-term plan after monsters are eradicated). His fate is non-guaranteed, and you can choose to kill him or let him live.
The Cats on the other hand have an associated questline in 3 emphasizing that their process is the most unstable, the most dangerous and produces the weakest results with no focus that have a habit of going into psycho bloodrages like a fox in a henhouse whenever they get stressed out (and they trade it off by being produced via a travelling caravan of kidnappers that go through dead orphans at an expendable rate). Cats live fast and die young, and while Geralt catches flak for getting involved in human affairs, most Cats are hired thieves and hitmen for hire (making them the most disliked and valid-to-kill of all the schools, in a career path that everyone dislikes anyways).
Cut to now.
The entire point of the Witcher 3 (and by extension, the entire narrative beforehand) is Geralt (the Witcher) and Yennefer (the Witch) trying to keep Ciri (their adoptive daughter) safe, since they were both made involuntarily sterile due to magic, and both just want a family.
Ciri is a daughter of the Elder Blood, which in laymens terms means she can hop dimensions and has access to a different type of primal magic. She can't cast flame spells or runic circles, rather she warps around, modifies her own speed on a time-control level and is a potential walking nuke. Naturally, many factions would like a walking nuke, especially one that can have kids who are also walking nukes (from the Elves, to the Lodge of Witches, to the Empire, and more); so they chase her to the ends of the continent to try to capture her / rape her before she learns how to control it. Geralt and Yennefer just want to live a family life with her, so when she goes missing, they both go on a fucking warpath and individually tear through the continent to find her. Geralt has a hairtrigger for beating men to death who even looked at her the wrong way
(at least after he recovers from amnesia in the Witcher 1-2 owing to other events), and Yennefer brags about torturing information out of people while maneuvering high politics alone just to find Ciri and keep her safe.
And yet, the basis of the Witcher 4 is that Ciri is now somehow a processed Witcher, dealing with peasant drama in Koviss.