Your question was why the advice was given to avoid ThreadLocals. This is the primary reason. It's not necessarily related to avoiding a larger memory footprint.
Typically ThreadLocals are used whenever something is costly to initialize per-request. The rule of thumb is usually: you might want to use some heavy object without locking, so you stick into a ThreadLocal.
However, if suddenly you have a million threads then this optimization doesn't work anymore. Sure, the concept of ThreadLocal still works, but in practice you'll end up creating a million of these heavy objects - something you wanted to avoid!