I feel like there are two questions here. (1) Does English language break down into metrical feet and (2) are metrical feet used in English poetry.
The American Poetry Foundation defines a foot as a measurement of accentual-syllabic meter, which is just a way of saying English speech regards stresses and syllables distinctly. The breaking up of large rhythmic structures into feet is natural. In analysis of poems with strongly regularized rhythm, feet are used to determine the normative meter.
We can apply them to Blake, which will comment on the Coleridge question, and highlight the potential ambiguity in metrical interpretation. Specifically, the second stanza of Blake's London can be delivered variously in conjunction with the normative Iambic Tetrameter of the poem:
In ev/ery cry / of ev/ery Man,
In ev/ery In/fants cry / of fear,
In ev/ery voice: / in ev/ery ban,
The mind-forg'd / manacles / I hear
(I broke up the final line into three feet for ease of reading, utilizing first a bacchius and second a dactyl to set up a final iamb, although it can certainly be done with four feet.)
Alternately, the stress on "forg'd" can be suppressed to result in a spondee on the final foot:
The mind-forg'd / manacles / I hear
Utilizing an amphibrach for the first foot. It seems less graceful to try to impose a four foot structure on the 4th line, and there is a case for a polyrhythmic approach because the last line can also be delivered with only three stresses, by suppressing the stress on the "I":
The mind-forg'd / manacles / I hear
Preference of the reciter is the driving motivator.
The more regularized a poem in English, the more apparent the feet will be.
Shakespeare is famous for iambic pentameter but it's not always fully regularized. One of my favorite pentameter poems begins with a trochees which morph to iambs:
Thou hast / made me, / and shall / thy work / decay?
and which I render entirely trochaic in recitation:
Thou hast / made me-- / shall thy / work de/cay?
In this Holy Sonnet of Donne's, iambic pentameter is the normative meter and strongly manifests itself to an English speaker after the first line. Breaking it in the 9th and 10th lines adds power to the poem, but requires analysis in terms of where the stresses might go. Thus feet are useful.
I think feet do appear even in modern poems:
I saw / the best minds / of my / generation / destroyed by / madness, / starving / hysterical / naked, / dragging /themselves / through the / negro / streets at dawn / looking / for an / angry / fix
but that it's more rarely applied as an analytic or compositional tool in free verse, as the previous line could be delivered with a different number of stresses, or different stress positions.
Eliot demonstrates that meter can still be strongly imposed on freer verse that resembles prose, save for the strong rhythm, making a case that meter is the fundamental distinction of poetry:
That was a / way of / putting it -
not very / satis/ factory:
A peri/phrastic / study / in a / worn-out / poetical / fashion,
Leaving / one still / with the / intolerable / wrestle
With words / and meanings.
Note that "intolerable" does not fit neatly into a tetrasyllable, having five syllables unless contracted, which is part of the tension Eliot is comenting on. Again, like Ginsburg, there are numerous way to break these lines into feet. (It's an art, not a science! ;)
But it is difficult to contextualize certain famous modern poems without a recognition of meter:
This Is / Just To / Say
I have eaten / the plums / that were in / the icebox
and which you were / probably / saving / for breakfast
Forgive me they / were delicious
so sweet and / so cold
How it is broken up into feet is less important than that there is a strong, underlying meter which renders this a poem as opposed to a mere note on the refrigerator.
The further we get from the Elizabethan period, the more unruly the phrasing becomes, and by the modern poems, definitive application of scansion is problematic because the poems rarely have a discernible normative meter.
But to demonstrate unequivocally that feet do exist, and have application even in modern English poetry I'll use a work from the poet widely regarded as the greatest of the 20th century:
Turning / and turning / in the / widening / gyre
The fal/con can/not hear / the fal/coner;
Again, my breakdown of the first line is in no way definitive, (and I'm not entirely happy with my choice, though the intent is readability and as a guide for delivery--it has been pointed out that a case can be made for a stress on the "in", which I personally feel is sub-optimal;) but it's indisputable that the second line is in perfect iambic pentameter.
The genius of this choice is a factor of the beats to syllables ratio between the first and second line--4/11 vs. 5/10. This creates an extraordinary effect that might be termed "metrical compression", and itself can be understood as the merging of traditional and modern English poetry.
Iambic pentameter might be taken as the normative meter in that in reappears in many important lines, and is used to set up or contrast the metrical variations of previous or subsequent lines. The other indisputably iambic pentameter lines is the poem are:
Mere an/archy / is loosed / upon / the world,
The best / lack all / convic/tion, while / the worst
The dark/ness drops / again; / but now / I know
And what / rough beast, / its hour / come round / at last,