You just need to wait a while. Sounds like you got a solid job/income but no history and you really got screwed with all those hard pulls. That stays on your report for 2 years. With no credit history you should have built up for a while instead of shooting for the big cards right away.

Also you should know the free credit scores you're getting are no good. Get a real FICO score. You'll have to pay for this or get a free trial of some kind. I use ID guard through Costco to get all 3 credit agency scores and 3 credit agency reports each month. There may be cheaper out there but I like this service. They also send alerts every time something happens on your report like hard inquiries, new accounts, etc.

Could be your actual FICO scores are lower than 700 which will definitely affect your ability to get a good rewards card. You should be at 740 minimum for the best cards, ideally 760-770 or higher.

Also see if you can convince a parent/relative/friend who has an Amex card to add you as an authorized user. You inherit their card history from the date they opened it and it goes on your credit report and shows as part of your available credit.

Your reports also show all hard inquiries so you know how many pulls you have for each agency. If you've been trying to get Chase and Amex cards you should have no pulls for Transunion unless they did secondary pulls from there when you called for reconsideration. So if you really want a card now your best bet may be to get an entry-level card from an issuer who pulls from Transunion.