When would the verb 'start' be used without an object?

No, there are just loads and loads of intransitive uses of start that mean something like to jump or startle. Indeed, that’s the oldest use of the verb, which is why they’re all under Branch I below.

Here are just a few of them, courtesy of the OED.

I. To (cause to) make a sudden movement, and related senses.

1. intransitive. To leap, jump; to caper, cavort; (also) to leap or spring upon a horse. Also figurative. Obsolete.
2. a. intransitive. To move with a bound or sudden impulse from a position of rest or repose; (also) to come suddenly from or out of a place of concealment.
   
   c. intransitive. To rise suddenly on or upon, or (in later use esp.) to, one's feet (formerly also occasionally with legs); to stand up.

4. a. intransitive. Of an inanimate object or substance, esp. a liquid: to issue suddenly and with force; to fly, flow, or be projected by a sudden impulse. Frequently with out, out of, from.

6. a. (a) intransitive. To make a sudden movement, esp. of part of one's body, as to avoid a blow or perceived threat; to flinch or recoil from something in alarm or repugnance. Chiefly with from or with adverbs (as aback, aside, away, back, etc.). Also with the part of the body as subject.
   
   (b) intransitive. Of a horse: to shy; to swerve suddenly from its course in fright or alarm; to bolt (also with adverbs, as aside, astray, etc.).
7. a. Hunting
   
   (b) intransitive. Of an animal: to emerge, esp. suddenly, from its lair or place of refuge. Also with up.
8. a. intransitive. To undergo a sudden involuntary movement of the body, resulting from surprise, fright, sudden pain, etc.; (sometimes without implication of actual movement) to feel startled or momentarily perturbed, as at a sudden realization.
9. a. (a) intransitive. To awake abruptly out of (also from, †out) sleep, a daze, etc.; to come awake with a start.
   
   b. intransitive. To come suddenly, or with a start, into (formerly †in) a particular state or condition; to fly into a rage, burst into life, etc.; (also) to go suddenly out of one's wits. Formerly also with †off. Now rare (in later use coloured by senses in branch II.).

There are many more than just these that I’ve selected here.

The Branch II senses that you’re thinking of mean to begin, but the Branch I senses are much much older, beginning way back in Old English when we used 3-digit years. :) The Branch II senses started in Early Modern English, and those are probably the only ones you are thinking of. Those don’t apply here.