There is no problem about the speed of transaction or the confirmation period whenever you send bitcoin to anyone.
It depends on how much your fee does.
That is definitely correct, the higher the fees you pay for a transaction to proceed the faster the transaction is especially these days when there is no network congestion happening in the Bitcoin network but if there are network congestions that will occur just like what happened way back in the year 2017 when the market is pretty bullish and there are a lot of people using Bitcoin it will surely cause some delay even if you use high fees on paying a transaction.
And also we already have Lightning network that make transactions easier and faster so it's not a problem anymore.
Congestion during the all time high was due to network spam and now, after experiencing that. We already have that solution that ease and make transactions faster even at the peakest moments.