Flash and Illustrator took very different approaches to the object-building workflow. What you're attempting to do isn't something you can do very easily in Illustrator.
You can still edit the paths with the open arrow tool after enabling live paint, but it's extremely finicky. You can sort-of mitigate some of the issues by screwing around with the live-selection tool and 'deleting' the chunks of area that were automatically defined when applying live paint, which (sometimes) causes adjacent chunks to merge. Other times it breaks things. Though, to be fair, I don't use Live Paint very frequently so my experience is probably much different than the experience of someone who knows how to use it correctly.
As @Billy Kerr mentioned in his comment, you may be able to achieve something similar to what you can do in Flash with the shape builder tool, but that merges the combined objects together into a single shape that can't be broken down into its original pieces. If you still want to be able to manipulate the constituent objects freely after merging them, you might want to look at using the tools available in the Pathfinder panel (Window > Pathfinder). The Shape Mode operations (first row of icons) are the tools you want to use here. They behave the same as the Shape Builder tool in that the objects will merge or subtract into a single shape. However, if you hold Option (alt on PC) while applying these operations, the shapes will instead merge as a compound object - allowing you to work with the constituent shapes as if they were grouped objects.
MUCH LATER EDIT:
So there's this plugin from AstuteGraphics that appears to do what you're looking for. It's called DynamicSketch. https://astutegraphics.com/plugins/dynamicsketch