Also any given node is not going to be changing its amount of BTC very often, at least for normal users, so if there are millions of LN nodes and your route takes your payment through a handful of nodes, the likelihood that any of them will have performed a transaction and fallen below the amount you are sending in that very small amount of time would be extremely rare. That at least covers nodes that aren't major hubs of activity. But nodes that are major hubs of activity are likely to have a lot of BTC stored in them so again that just won't be a problem.

Also remember that even VISA, the largest transaction processor in the world, only averages like 3000 or 4000 txs/sec. So while the LN may be able to technically handle on the order of 1 million txs/sec, it will never get close to that number, which bring the likelihood of a node on your route making a payment in the time between your payment's route calculation and sending it even that much less likely. This still might occasionally occur, in which case maybe the payment would just take a few extra seconds to go through due to needing to calculate a new route and resend the payment.