Combine crumble ingredients by cutting butter into the flour and sugar until pea-like crumbs form. Spread on a baking sheet and bake at 375 degrees F for 10-12 minutes. Cool then crumble.
Combine rhubarb, sugar, and lemon juice in a shallow pan. Bake for 30-35 minutes until soft and syrupy. Cool, then blend. Cool completely, then chill in the fridge until ready to use.
Whisk cornstarch with ¼ cup of the milk in a small bowl to create a slurry. Set aside.
Heat cream, remaining milk, sugar, corn syrup, and salt in a medium saucepan over medium-high heat. Bring to a boil, being careful to make sure it doesn't boil over, and cook for 4 minutes. Whisk in slurry and cook for another 2 minutes to thicken.
Pour about ¼ cup over the cream cheese and whisk to combine until smooth. Add remaining milk mixture. Stir in vanilla extract.
Make an ice bath with water and ice. Pour the ice cream base into a bowl and nestle it into the ice bath to chill quickly or chill in the fridge overnight.
When thoroughly chilled, pour the ice cream base into the bowl of an ice cream maker and churn until it is the consistency of soft-serve ice cream. Add all but ½ cup of the crumble pieces during the last minute of churning.
Layer half of the ice cream with half of the jam, then repeat with the remaining ice cream and jam. Use a knife to swirl the mixture a few times, then sprinkle the reserved crumble pieces on top. Set in the freezer for 4-6 hours to harden.