Think of all the other offers that you will definitely miss out on if you keep passing on other cards due to 5/24. A bird in the hand is better than two in the bush and all that. Don’t go for any old offer but there are plenty of good offers available from other card issuers at any random point in time. Make sure you have a long term strategy for using the miles though so that you don't end up with a bunch of miles in a program that for whatever reason your may never use, or never have quite enough miles to get anything worthwhile from them. Also make sure you have a strategy to ensure the miles do not expire, fairly easy and inexpensive to do if you research each program.

If you max out 5/24 solely on Chase cards then you will have most of the decent ones anyway. The Sapphire (Reserved or Preferred) now requires you to wait 4 years between applications so the cards from other issuers will fall off your credit report long before you become eligible for that one again. Most of the others require two years after you receive the bonus before you can apply again. If you apply for cards from other issuers shortly after you get 5 Chase cards they will also fall off your credit report by the time you are eligible to re apply for another Chase card. Remember you can prepay a lot of bills if you are struggling to make the spend requirement. I have been known to prepay my utilities up to 6 months in the future to meet a spend requirement on time.