It's a math problem which only you can answer. First estimate your average checking account balance over the year and multiply it by the account's interest rate, then estiamte the amount of spending that you could put on a credit card over the year and multiply it by the credit card return rate. Go with whichever gives you more money in a year.
Lets say you have $10k average in checking at 3% and $20k average spend which could be at 2% on a credit card. You would get $10k*.03=$300 from checking and $20k*.02=$400 from the card in this example, so the credit card would be the better choice. You should not accept less than 2% return on a credit card, and probably higher.
Consider Alliant Credit Union (no personal experience, have just seen them online). They have an 0.65% checking account with no other requirements, and offer a 2.5% cash back credit card with $59 annual fee, possibly the best single-stop cash back credit card and no-hassle checking combination. With the above numbers you get $10k*0.0065 + $20k*.025-$59 = $506. So if you are not getting a better result with your independent checking and CC, you should consider changing to Alliant. You might also find a better credit card to pair with Schwab's popular 0.35% checking account.
