The Shulchan Arukh's stipulation that only insects visible to the naked eye are prohibited (84:36) carries the suggestion that checking foods visually is halachically necessary.
This document brings several sources that the checking process itself ("bedikas hamazon" or "bedikas toalim") is indeed halachically required:
What frequency level of infestation obligates a person to check a
particular fruit or vegetable? If it is a “miut ha’matzui”, a frequent
minority, the vegetable must be checked. What percentage is considered
a “miut ha’matzui”? This is the subject of dispute between many
authorities. Rav Moshe Vayeh in his sefer Bedikas HaMazon K’Halacha
(Part II 3:2 footnote 3) quotes the Responsa Rivash 191 that it must
occur with a frequency of close to 50%. The Mishkinos Yaakov (YD 17)
says that there is an obligation to check for insects even if they
occur ten percent of the time. The Shevet HaLevi (IV:81) and others
explain that the b’dika is not based on a specific percentage; rather
there is an obligation to check any item which is frequently infested
by insects. Even if the particular item has a low percentage of
infestation, if we see that it occurs regularly, there is an
obligation for bedika. Rav Shlomo Z. Auerbach rules like the opinion
of the Mishkinos Yaakov and that the number 10% is determined by the
item in question. For example, if one of ten (10) heads of lettuce
contains one or more insects, it requires bedika. NOTE: If an item
which does not require inspection was, nevertheless, found to contain
three or more insects, it must be fully inspected. If inspection is
not possible the food must be discarded (Shulchan Aruch YD 84:9).
According to HaGaon Rav Shlomo Zalman Auerbach ZT’L, produce which
cannot be checked [e.g. broccoli and cauliflower florets, frozen
spinach, canned asparagus] may be used without inspection in the
following manner: Break apart florets, agitate and soak in water for
ten minutes, pour off water, cook until soft. Puree finely in a
blender or food processor. Use as desired (e.g. in kugels or quiche).
This text seems to imply that checking is both a halachic requirement under certain circumstances and optional under others.
Kof-K does not, however, supply any halachic sources for the instructions they give for checking, and I do not remember seeing such citations in any other guide. So there is still the question of who decides what checking procedures are acceptable, how, and whether this decision is based on halacha or mere practicality.