Is continuity just causation?

Generally yes, but the simile might be not the best, since spark can easy be taken as beginning point (which drives one back to confusions of "science") and it's causality of a wheel might not expected. So better take fuel or nourishment of an long ongoing fire here, the wandering on, a point of beginning can not be seen.

By uprooting the cause, the effect does not arise. By, or with the, total uprooting, destructing of this very avijja, not-knowing, ignorance, without remaining, birth, aging, sickness and death, all the suffering... comes to an end.

(Not given for trade, exchange, stacks or other wheel-binding undertakings and entertainments, fuel, but as a means for lighten and release only, an end of continuing)

Dependent origination (paṭiccasamuppāda) does not mean "causation"; even though dependent origination is an example of "causation"; just as a "poodle" is an example of a "dog". In Pali, the word for "causation" is idappaccayatā.

A fire from a spark is a purely physical causal phenomenon. Whatever continuance of the original spark that exists in the fire is the subject matter of the laws of physics; which are idappaccayatā but are not paṭiccasamuppāda.

Paṭiccasamuppāda definitively refers to twelve mental conditions that result in the origination of mental suffering (dukkha). The Pali scriptures say:

And what is dependent co-arising? From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes mind-&-body. From mind-&-bodyas a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes attachment. From attachment as a requisite condition comes becoming. From becoming as a requisite condition comes (self-identity) birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering.

Since "continuance" is the impression of a period of "permanence", it appears unrelated to "causation", which refers to the causes & conditions that result in "continuance" rather than "continuance" itself.

Two Pali words for "continuance" are "ṭhiti" & "ṭhita" (both apparently derived from "tiṭṭhati"). In the most extreme permanent case, ""ṭhita" is used for the eternal nature of the Laws of Nature, which are permanent (for example, in SN 12.20 about the law of how suffering arises and AN 3.136 about the three characteristics).

Below from chapter SN 26 is a (rarely discussed) modified teaching about the dependent origination of suffering, which includes "continuance".

Yo kho tejodhātuyā uppādo ṭhiti abhinibbatti pātubhāvo, dukkhasseso uppādo, rogānaṃ ṭhiti, jarāmaraṇassa pātubhāvo

The arising, continuation (ṭhiti), production and manifestation of the fire element, is the arising of suffering, the continuation of disease and the manifestation of old age and death.

SN 26.9

Most Buddhists would interpret the verse above as referring to the arising of physical phenomena however the meaning is mental. All of the relevant words, such as "production" and "manifestation" are generally used for mental phenomena.

Thus, the manifestation, production, arising and continuance of the fire element means, for example, the mind gaining a period of pleasure from eating a delicious hot food and producing self-clinging or self-identity from this attachment & delight towards this delicious warming hot food.

Or enjoying & indulging in lying in the hot sun or in a hot bath; or complaining about hot weather or screaming about a burn on the skin. This is the meaning of arising, continuation, production and manifestation of the fire element.

The arising, continuation, production and manifestation of the fire element does not refer to a bush fire caused by the sun or lava flowing from a volcano.

It refers to the dependant origination of craving, attachment, self-becoming and suffering towards the fire element. The suffering occurs due to the aging-&-death of the self-identifying attached to that fire element.

When we speak about causation we're used to thinking in terms of discrete entities interacting and causing distinct events. This is normal in materialistic view, ever obsessed with discreteness. However, when we speak about continuity, we are in the analogue world - where everything morphs and flows, and where individual "things" and "events" are more like the sums of the amplitudes of multiple intersecting waves rather than truly isolated entities.

Also, in Dependent Origination we are not as much talking about causation as we're talking about what Charles Sanders Peirce called "the habit formation", or the tendency of tendencies to self-perpetuate and grow.

You know how in men toilets there is a self-perpetuating tendency for the floor under the urinal to get wet? (It self-perpetuates by causing men to stand further away.) The same exact principle is behind Dependent Origination, and, according to Peirce, behind Evolution at large.

Mind is integrated information, regardless of media. Accumulation of regular patterns leads to development of mind's faculty of recognition, and mind's tendency to form habits and to seek comfort in familiar experiences. This in turns leads to emergence of purposeful activity towards comfortable experiences. This tendency self-perpetuates and develops into a full-blown Umwelt, complete with "objects", "other living beings" and "I".

So yeah, continuity and causation are two alternative descriptions of essentially same thing, but the tendency for habit formation is the real key in D.O.