Worth separating the clocks, because my payout is slow covers three unrelated mechanisms and the fix for each is different.
1. The payout processor interval
The payout plugin does not broadcast the moment you approve. It collects approved payouts and processes them on a timer. On a default install that timer is long enough that a payout approved just after a cycle waits most of an hour before anything touches the network. This is deliberate — it exists so five payouts to five addresses become one transaction — and it is the single most common cause of nothing appearing to happen.
2. Fee estimation
Your node estimates a fee from what it has seen. A node that was offline, or one with a small mempool cap, has seen a narrow slice and will happily hand you a rate that was correct four hours ago. You sign at that rate, the transaction sits, and you conclude the payout is stuck when in fact it is broadcast and unattractive.
3. The wallet's own coin selection
If your available UTXOs are one large one and forty dust ones from Lightning closes, the wallet may build something enormous, and enormous transactions at a marginal fee rate wait. I have watched a 4 kB payout sit through six blocks that were clearing everything above its rate.
Only the second and third are worth engineering around. The first is a setting.
What I have not tested: whether the interval behaves the same on the Lightning payout path. My node has almost no LN volume so I have no useful sample and I would not guess.
Short version: check the processor interval before you touch anything else, then check what your node thinks the fee market looks like, and only then start blaming the network.