Full story Machinima style series are rare, but if you just want heavy editing and a somewhat coherent plot as opposed to nearly unedited gameplay, Alpharad and LarsBurrito might work. Alpharad heavily edits his videos and usually writes a script to go over the gameplay that does a good job pulling a story out of the footage. LarsBurrito does a similar style, but also often does themed playthroughs where he writes the script to flavor the playthrough to fit whatever character he's roleplaying as.
If you want actual story but are ok with significantly less editing, Mianite is a series I rewatch every once in a while in a similar way you describe. The scripted story doesn't really start picking up until a significant way through season 1, but there is still enough conflict between the different players to make it more than just a Let's Play.
VeggieTales is giving them too much credit - there's good moral lessons in there! (And the creator said all the botanically-fruit male veggies are canonically trans??)