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If we look at this "sound family" of words, we see that they all seem to be about things going wrong:
crumble, tumble, rumble, fumble, bumble
(Perhaps there are others, although I exclude humble, mumble, grumble as not very similar.)
Is there an etymological connection here — for example, some kind of pejorative suffix in older forms of English or the source languages — or is it coincidence?
I don't know anything about their etymological relationship; however, they all seem have a common 'phonaestheme' -umble which is associated with failures/ things going wrong (as the OP pointed out).
A phonaestheme is a particular sound sequence that suggests a certain meaning, for instance, words beginning with gl- are often associated with light reflection or vision as in glitter, glisten, glow, gleam, glare, glint, glimmer, gloss etc. The study of phonaesthemes is called phonaesthetics (or sound symbolism). (I have explained phonaesthemes in another answer to a question asking about the gl-.)
According to this article (Sound Symbolism in English: Weighing the Evidence by Barry J. Blake) on Tandfonline, most words with the sequence -umble can be said to refer to lack of success such as bumble, crumble, fumble, jumble, stumble, tumble etc.
Also from How to spell the meaning of words by Trevor Lloyd, the -umble words, bumble, crumble, fumble, grumble, humble, jumble, mumble, rumble, stumble, tumble have a common semantic theme of mild disparagement.
According to Complex Words in English, -umble also links words having to do with noise: grumble, rumble, shumble, stumble, but fumble, mumble, stumble and tumble suggest 'repeated awkward movements'.
