You are quite right on both counts. You have your neurology correct and you’ve come to the right conclusion. A bulging disc pressing against the L5, S1 nerve root wouldn’t cause quadriceps or hamstring muscle weakness.
Leg weakness (specifically around the knees) on both sides can occur when there’s a central bulge at a higher level (around L2 to L4) as you pointed out.
A recent study was done on the status of trunk and leg strength in patients with lumbar disc herniation (LDH). The researchers concluded that pain, not nerve root compression, is the reason weakness occurs.
Muscle strength may also be affected by fear of movement causing pain. The person avoids any movements that might cause pain. Eventually the muscles controlling that movement get flabby and atrophy. Weakness is the natural result. Kudos to you for thinking through this problem!