There is perhaps no symbolic significance to this method.
I wondered why the head would be removed this way, rendering the meat unfit for consumption.
The Judaism.SE site offered one explanation for this unusual (unkosher) method of slaughter:
Why is melika - not shechita - done on the bird korban? - Mi Yodeya.
The Sefer HaChinuch (13th century) explains that doves are offered only by the poor.
Generally these people have little time for ceremony and must return to their work as soon as possible, so the possibly lengthy slaughtering was replaced by a much quicker method (humane, but not kosher).
Presumably, since the offerings were burned entirely and not for consumption, God would understand the reason and accept them.
Similarly, in Leviticus 5:7–11, two birds are offered:
But if one’s means do not suffice for a sheep, that person shall bring to יהוה, as the penalty for that of which one is guilty, two turtledoves or two pigeons — one for a sin offering and the other for a burnt offering.
The offerer shall bring them to the priest, who shall offer first the bird for the sin offering, pinching its head at the nape without severing it.
He shall sprinkle some of the blood of the sin offering on the side of the altar, and what remains of the blood shall be drained out at the base of the altar; it is a sin offering.
And the second bird he shall prepare as a burnt offering, according to regulation.
For the sin of which one is guilty, the priest shall thus make expiation on behalf of that person, who shall be forgiven.
— Leviticus 5:7-10 (JPS)
Notice that while the second bird was to be killed as in Leviticus 1, the second bird, to be eaten and not burned, is killed without totally severing the head.
The NASB20 translation describes this process more clearly:
He shall bring them to the priest, who shall first offer that which is for the sin offering, and shall pinch off its head at the front of its neck, but he shall not sever it.
This isn't the full kosher ceremony, but it approximates it, using the fingers in place of a knife.