Pre-approval offers are marketing, not underwriting. The fact that you were pre-approved for a $500 store card doesn't necessarily mean that's all Synchrony would approve if you actually applied (could even be a denial). Likewise, your recent Amex approvals don't obligate another lender to reach the same conclusion. Every lender has its own underwriting criteria.
As for store-only cards... my personal opinion is that I'd pass, especially with a young or thin file. Their usefulness is limited since they can only be used at one retailer, and if you have any plans of building a relationship with Chase, you've also used up one of your 5/24 slots on an account with fairly limited long-term value. I'd rather save the application for a general purpose Visa, Mastercard, etc...
