What a lovely story and what a dedicated custodian you've been to this little oak for four or five years! A true Garden Ninja!
The fact that you've taken such care over the repotting process, gently untangling the roots rather than just hacking at them, already puts you well ahead of most people in this situation. Let's work through what's happening and what you can do to give it the best chance going forward.
What the Browning Leaves Are Telling You
The pale brown new leaves and the dry, darker patches on the older foliage are almost certainly the result of two things happening at the same time. After four or five years in a milk carton, the tree will have been severely root bound, meaning the roots had completely exhausted the available space and were circling and compressing themselves rather than being able to spread and function properly.
A root bound tree cannot take up water and nutrients efficiently even when both are present, which is why you get those classic stress symptoms on the leaves despite the tree being watered regularly. Combine that with the nutrient exhaustion that comes from years in the same small volume of compost, and the leaf browning makes complete sense. The good news is that you've already addressed the root cause by repotting it.
Those Little Webs
The small webs you noticed are almost certainly spider mites, which thrive on stressed plants and in warm, dry conditions. Los Angeles in a hot spell is exactly the environment they love. They're not a death sentence but they are a sign the tree was under stress and they can accelerate leaf damage if left unchecked. Give the leaves a gentle wipe with a damp cloth and if you see them returning, a very dilute solution of washing up liquid in water misted onto the leaves will deal with them without harming the tree. Keep an eye on the undersides of the leaves in particular as that's where they tend to congregate.
The Repotting You've Done
You've done well here. Gently extending and untangling the roots rather than cutting them, soaking the new compost before planting, and moving it into a much deeper container are all the right calls. The taproots you identified in the centre are crucial for the long-term stability and health of the tree, and the fact that they were firm and straight is a very good sign.
One thing worth adding to the new potting mix going forward is a slow release granular fertiliser worked into the top layer of compost, as potting soil alone is relatively low in nutrients and an oak that has been starved for a while will benefit enormously from a gentle feed once it has had a couple of weeks to settle into its new home. Don't feed it immediately as the roots need time to adjust first.
Aftercare Right Now
Keep it out of direct harsh afternoon sun for the next few weeks while it recovers from the repotting. A bright spot with morning light and shade from the most intense afternoon sun in Los Angeles would be ideal during the recovery period. Water when the top inch or two of compost feels dry, and make sure the container has good drainage so it never sits in waterlogged soil. The leaves that have already browned won't recover but watch for new growth emerging, which will tell you the tree is settling in well.
A Gentle Word About the Future
I want to be honest with you because I think you deserve a realistic picture of where this little oak is heading. Oak trees are magnificent but they are not really suited to life in a container long term. A native oak, whether that's a North American species or a European one, is one of the largest and longest lived trees on the planet. A mature oak typically reaches 20 to 40 metres in height with a canopy spread of a similar scale, and the taproot system alone can extend many metres deep into the ground. In a container, even a generous one, the tree will eventually reach the limits of what the pot can support, and you'll face this same situation again within a few years on a larger scale.
The kindest long-term outcome for your oak would be finding it a permanent home in the ground, whether that's your own garden, a local park, a community green space, or through a tree planting charity or scheme. Many organisations in the US actively welcome donated native tree saplings. That doesn't need to happen today or even this year, but it's worth thinking about as the tree continues to grow.
For now, though, you've done a genuinely good job with a difficult situation, and this little oak is lucky to have someone who cares about it as much as you clearly do. Keep a close eye on it over the next few weeks and do post an update when the new growth comes through.
Lee Garden Ninja