A clause always has a verb, but that verb may be of two sorts:
- a finite verb clause: one inflected for person, number, and tense
- a non-finite verb clause: one that is NOT inflected for person, number, and tense
The clause further contains all the arguments for a verb. However, the subject argument is optional in non-finite verb clauses. These are the kind that you may have learned to call “verb phrases” rather than “verb clauses”.
Sometimes non-finite clauses do have subjects. Here’s an infinitive clause with a subject:
- For John to quit now would be a scandal.
That uses a for-complementizer to give to quit a subject of John. We keep the complementizer but insert a dummy-it when reversing that copula:
- It would be a scandal for John to quit now.
With -ing verbs, no for-complementizer is needed, nor allowed:
- John quitting now would be a scandal.
When your non-finite clause is a gerund–participle (GP) one, you can also use a possessive for the logical subject of the non-finite clause. These are both grammatical:
- I see John quitting now as completely scandalous.
- I see John’s quitting now as completely scandalous.
In your own example:
Walking to school, I spotted a car driving erratically toward me.
You have a gerund-participle (GP) clause that’s acting as some sort of adjunct. It’s an adjunct, not an argument. You can remove it without “breaking” your sentence.
Under one possible analysis, your GP clause would be considered an adnominal adjunct, one modifying only the subject. (It’s probably easier to call these by their shorter name, noun adjuncts.)
However, under another possible analysis your GP clause is an adverbial adjunct, one that applies to the predicate or the entire sentence as a whole.
A different kind of participle (the past/passive one, not the present/active one) can be used to form a nominative absolute:
- Breakfast finished, I hurried off to school.
Your GP clause is not an absolute construction the way that freestanding noun plus participle is in the example I just gave, but exactly which sort of adjunct you choose to call it — adnominal or adverbial — can be argued either way. And often is.
Consider these variants:
- Luckily, I spotted a car driving erratically toward me.
- Walking to school, I spotted a car driving erratically toward me.
- Ten minutes later, I spotted a car driving erratically toward me.
- Tuesday morning, I spotted a car driving erratically toward me.
- Early this morning, I spotted a car driving erratically toward me.
- Along my morning walk, I spotted a car driving erratically toward me.
To my mind those are all structurally similar in that they have some sort of adverbial phrase at their start, one which applies to the entire sentence following it. For your GP case, that’s easily converted into a normal prepositional phrase or subordinate clause:
- Upon walking to school, I spotted a car driving erratically toward me.
- When walking to school, I spotted a car driving erratically toward me.
- While walking to school, I spotted a car driving erratically toward me.
- While I was walking to school, I spotted a car driving erratically toward me.
