Where are you getting 5% cash back? Highest I've seen is Chase Sapphire with their points which are worth 2-3%. Highest straight cash back I've seen is Citi DoubleCash at 2%.
Discover has seasonal 5% categories that change every 3 months but they're pretty broad like "Gas and Groceries" or "Restaurants and Entertainment".
I have the old American Express Blue Cash card which has some crazy 5% cashback on gas and groceries once you hit a minimum spend during the year. That thing paid out like a slot machine when gas prices were hovering around $3.50 a gallon and I was doing a 100 mile commute daily.
Sure, fees are under 5%, but what about that one month that you didn't pay off your bill in full? Enjoy that ~30% interest rate on your account balance.
1) Chase Freedom's APR tends to be around 20%
2) If you only carry a balance (or part of it) for one month (as in your example) and then pay it off the next month, your interest charge is nowhere close to the APR (and it's only calculated against the portion you didn't pay off).
I have a discontinued Barclaycard that gets me 5% for gas, groceries (including Walmart and Target), and bookstores (including anything at Amazon, even AWS and digital downloads). There are caps, but I almost always spend below (have hit the grocery one a couple of times).
In addition to the ones mentioned here Wells Fargo has a card with 5% back at grocery stores for the first 6 months and the Chase Ink+ gets 5x at office supply stores and a few others.
USBank Cash+ too. You have to pick categories each quarter. You get to pick though, which makes it more convenient than cards that just assign random categories.
I have the old American Express Blue Cash card which has some crazy 5% cashback on gas and groceries once you hit a minimum spend during the year. That thing paid out like a slot machine when gas prices were hovering around $3.50 a gallon and I was doing a 100 mile commute daily.