It's ok to send it to an Exchange, well I'm my case it is Volabit ( not sure if it's considered an exchange). I just let the BTC accumulate there until I get like a 0.5 or 1 BTC and then I transfer them to the hardware wallet (if I don't sell them right there and then). I think it is completely fine just don't keep too much BTC there, I personally don't let it go over 1 BTC. For me Volabit works the best, I like the page layout, notifications and they charge me barely anything for pulling my BTC out. I've never had a problem with any of the transfers from them, and it Has been always very quick for me! 
I'm a little confused, why don't you just have the mining rewards go straight to your hardware wallet?
-Tamerz
That is what we were talking about. I have, had, all my winnings for BTC going to my Ledger Nano S. After that I usually sent the BTC to the exchange, but my transaction fee kept going up. The last time I got $60 or so bucks in coins (4-5 payouts worth for me) I went to send it to the exchange and the Nano wanted over $15 to send $60 bucks worth of bitcoin! (And that was on the cheapest Low Priority one) So I did a custom one and selected 100 Satashoi's but that ended up beeing too low and no miner picked it up to confirm it so it was stuck. So I used a transaction accelerator and it worked.
But ultimately the issue is, where to send mined rewards from here to that is the safest & cheapest to either keep or use on an exchange. He was asking why I don't have my rewards sent to my exchange and I was saying that everything I have read said not to have rewards sent to exchanges but to a wallet first, then to an exchange. Even the moderators at the exchange I use said to do it that way.
So that is the issue