The only one I have on the top of my mind is “Se mettre sur son trente-et-un.” or “Être sur son trente-et-un”

Which mean

Mettre ses plus beaux habits. Être très bien habillé.

note: I will update it later if I think about anything else.

There does not seem to be a widely known precept for this idea in French; it has to be translated more or less literally. The following way to put it appears to fulfil the need for a basic, faithful rendering.

• Une bonne tenue vestimentaire commence par les chaussures.

En fonction de l’explication davantage que de l’expression, sur le modèle de « l’habit ne fait pas le moine », influencé peut-être aussi par l’idée de « make or break (the outfit) », je dirais :

Les souliers (chaussures) font l’habit.

C’est à ses chaussures que l’on reconnaît l’homme / la femme de goût.

I don’t know the origins or level of solemnity of the original Japanese expression, but it seems that at least some of the YouTube titles linked in your question are using it to attempt to influence people to consider [trying/buying] various footwear “looks,” so maybe it wouldn’t be totally inappropriate or overly crass to consider expressions/slogans/quotes used in the fashion industry to convey similar notions [solely/primarily] in order to sell footwear to the public, such as:

On n’accorde jamais trop d’importance au choix de ses chaussures. …
[second sentence of the full quote, attributed here to Christian
Dior,
omitted to attempt to render it more gender-neutral and less
stereotypically offensive] … .
(via chausseurdepuis1885.fr)

or

Les chaussures peuvent faire ou défaire une tenue entière.
(from 1001chaussures.com)

As accurately noted by the OP in comments below this answer:

“[T]he phrase ‘(quelque chose) peut faire ou défaire une tenue’ …
is just as likely to be used to point out the risk associated with
adding an item seen as somewhat extra / secondary such as an
accessory [like jewelry] to your already sufficiently balanced look
…[as it is likely to be used to refer to] something fundamentally
important like shoes.”

With that good point in mind, I’m thinking that perhaps the focus of the phrase at issue could be slightly (and hopefully sufficiently) narrowed to something fundamentally important like shoes by:
(1) personalizing the phrase with the use of possessive articles instead of definite and indefinite ones and/or
(2) using forms of the verbs “faire” and “défaire” that do not include or require the verb “pouvoir” and the hedging/indefinite/”less-than-fundamental” notion that that verb might imply:

(1) Vos chaussures peuvent faire ou défaire votre tenue.
(from karlandmax.com, first sentence)

(2)Les/Vos chaussures font ou défont une/votre tenue.
(from chamaripaelevatorshoes.com, first sentence)

Les/Vos chaussures … sont le détail …qui complète une/votre tenue et qui fait ou défait un/votre look.
(from scenoscope.fr, first sentence)

(Please note that this “make or break” sense of “faire ou défaire” could possibly be calqued from English [as in “Shoes (can) make or break an outfit”], but to the extent that such a thing would matter, this 1791 use of the French version makes me think that, if anything, it’s just the opposite [from L’ Ami des patriotes ou le défenseur de la révolution, Volume 1, Issues 1-16, via GoogleBooks].)