Triage: what kind of damage is this
A chip on an outer rim is cosmetic and usually leaves the piece usable. A crack through the wall will propagate under thermal cycling and the piece will eventually fail, often while full. Crazing across a glaze is a surface condition rather than a break. A chip in an enamel lining exposes bare steel or iron, which then rusts and lifts the surrounding enamel — that one is progressive and the vessel is on a clock. A broken lid is the most common failure and the hardest to solve, because lids are made to a body and a replacement rarely seats.
What kintsugi actually is
Repeated very widely and not evidenced. Recorded because the story is itself part of the culture, not because it happened.
Kintsugi is a Japanese repair method in which breaks are joined with urushi lacquer and the seams are then dusted with gold, silver or brass powder so the repair is emphasised rather than concealed. The practice is conventionally dated to the fifteenth century and is popularly linked to a shogun whose broken tea bowl was returned from China with metal staples; that account is repeated everywhere and TeaHQ has found no verification for it, so it is recorded here as a story the practice tells about itself rather than as history. Urushi hardens slowly over weeks and is what makes the join durable; the metal is decoration on top of it.
Kintsugi kits are mostly not kintsugi
The kits widely sold under the name use epoxy or synthetic resin with metallic powder rather than urushi. That is a reasonable way to make an attractive repair and it is a different material with different properties. If a repaired vessel is going to hold hot drinks, the question to ask about any adhesive is whether the manufacturer states it is suitable for that use and at what temperature — and for most craft epoxies the answer is that it is not stated, which should be read as no.
Other repairs, and their limits
Staple repair — drilling and pinning a crack with metal cramps — is an old technique found on Chinese and European ceramics alike and produces a strong, visibly mended piece. Re-tinning a copper vessel is a living trade in some places. Rust in an unlined iron kettle can sometimes be stabilised by boiling it out and re-establishing the interior layer, and sometimes cannot. Enamel cannot be repaired domestically at all. A ceramic filter that has been chipped is finished as a filter, though the pot may still serve with a strainer.
When to stop using something
A crack through the wall of a vessel that gets filled with boiling water is the case for retiring it, because the failure is sudden and the contents are scalding. A chipped rim on a drinking vessel is the case for retiring that vessel. Beyond those, a repaired pot that holds water and pours is a repaired pot that works, and there is a long tradition in several tea cultures of valuing a mended object more highly than an unblemished one.