> 2. The problem with that is that you may have infinite recursion, but you can only pass finite amounts of memory between recursion steps. This has also been addressed by https://cs.stackexchange.com/a/60978.

If I have two stacks (by spawning another thread), don't I have two PDAs ? That's what that stackoverflow post indicates, and IIRC two PDAs are equivalent to a Turing machine.