No it doesn't teach that at all. Ezekiel 18 clearly explains that children do not inherit the sins of their fathers. Salvation is also predicated on having understanding to understand the Gospel and obey it by being baptized. An infant can not do that on his/her own. Such "...children are born in sin." is false and should not be taught.
Tell you what: you better take some scissors to your Bible then:
Surely I was sinful at birth, sinful from the time my mother conceived me. (Psalm 51:5)
Even from birth the wicked go astray; from the womb they are wayward, spreading lies. (Psalm 58:3)
In case you haven't noticed, the most self-centered people on earth actually tend to be babies & toddlers.
Your family's toddlers didn't "environmentally" appropriate sinful habits.
Jesus talked about "sin" in the singular. “Very truly I tell you, everyone who sins is a slave to sin
The sins are the individual acts & omissions (wrong-doing or right-neglecting). Sin itself is a power that controls.
Infants & toddlers haven't wrung up the sins plural. But they inherit that power of sin in their nature.
That is standard Christian teaching across about all Christian denominations.
This is a denial of original sin and therefore a heinous error. It cannot be disputed that the sin of Adam has destroyed his progeny. I like a'Brekel's response to your verse, so I'll let him deal with you:
"Answer: Deut 24:16 is a law which God has given to man. From this we may not draw a conclusion as far as divine justice is concerned. The text refers to violations of the law and not to a breach of covenant. The one is not a necessary consequence of the other. The text refers to the sins of specific individuals. Adam, however, was the head of the covenant which was established in him with the entire human race. This sin was the sin of the entire human race, for outside of Adam and Eve there were no other human beings. The entire human race was comprehended in Adam, and thus that same human race bears the punishment of their own sin. Ezek 18:20 also speaks of specific sins of specific people, and is therefore not applicable to Adam and his descendants who are in covenant relationship with him. The text refers to adult children who do not follow the footsteps of their parents. God convinced them that they themselves were committing these sins, and thus would be punished for their own sins with the same manner of punishment. It is incontrovertible that God punishes children for the sins of their parents, as is to be observed in the flood, in the destruction of Sodom and Gomorrah, and in the children of Eli. God very expressly states the following about Himself: “... visiting the iniquity of the fathers upon the children unto the third and fourth generation” (Exod 20:5).