Noah and his family were the only ones left with purity of heart (they worshiped God) and pure (human) genetics. The Great Flood was not "starting over" but a cleansing of vile corruption that could not be allowed to continue.
The salvation provided by the ark did nothing to remove sin from Noah and his family. No where is there any hint that such a cleansing took place.
In answer to @curiousdannii's question, "how do you know that Noah and his family worshiped God?"
Gen 6: 8-9
Noah, however, found favor in the sight of the Lord.
These are the family records of Noah. Noah was a righteous man,
blameless among his contemporaries; Noah walked with God.
- Noah found favor in the sight of the Lord -- one cannot find favor in the sight of the Lord unless one puts the Lord at the center of his own life (or worships Him).
- Noah was a righteous man -- by all accounts, this would indicate he worshiped the Lord.
- blameless among his contemporaries -- the New KJV for v. 9:
This is the genealogy of Noah. Noah was a just man, perfect in his
generations. Noah walked with God.
... we can understand the implications of "blameless among his contemporaries". In other words, Noah was not perfect, but his
geneology was uncontaminated.
- Noah walked with God -- if not specific enough in the previous sentences contained in verse 9, this explicitly describes a relationship with God.
Now, why would I suggest that Noah's family (not just Noah) worshiped God? If a man - husband and/or father - has a relationship with God, then he is deemed the 'high priest' and directs his family in the ways of the Lord. Regardless of the sins recorded later by his sons, Noah's family did worship God.