The personal version of the Hyatt credit cards provides a small status boost. Along with the annual fee night certificate (and the opportunity to spend $15,000 for a second certificate), cardholders get 5 elite night credits each year and automatic low-level Discoverist status. But this only entitles you to a 10% bonus on points from stays, a free bottle of water, 2 PM late checkout, and the possibility of a minor room upgrade when available.
You can earn additional free nights by spending on the Hyatt credit cards, but it would take a lot of spending to get all the way to Globalist status. For every $5,000 you spend on the personal version, you'll receive an additional 2 elite night credits. If you decide to spend $15,000 with the card to earn the second free night certificate, you'd earn an extra 6 elite night credits, bringing you up to 11 credits (without counting your actual stays). For every $10,000 you spend with the business version, you'll receive an additional 5 elite nights.
If you wanted to get all the way to Globalist, without relying on any actual stays, you'd need to spend $140,000 on the personal version or $120,000 on the business version over the course of the the year. Even mid-level Explorist status, which doesn't come with fantastic benefits, requires 25 extra nights or up to $65,000 in credit card spending.
Hyatt doesn't usually match status from other hotel companies, except for a special partnership they have with the M life (MGM family) casino loyalty program. If you have Gold status (or better) with M life, you can automatically match it to mid-level Hyatt Explorist status. Unfortunately, no level of M life status is good enough to match to Globalist status.
High-level American Airlines elites can periodically earn Hyatt status through a "fast track" challenge program. Platinum and higher elites can stay just 10 nights in 90 days to earn Explorist status and Executive Platinum elites also have an opportunity to stay 20 nights in 90 days to earn Globalist status (elite nights earned from the credit card don't count). For many people, this isn't a realistic path to status.
A challenge with identical terms is sometimes available to employees of some large companies.
If you ever qualify for mid-tier Explorist status, you can hold onto it indefinitely. Hyatt's status year extends past the end of the calendar year. As a result, you can match your Hyatt status over to M life before it expires. Then you can match back from M life. Explorist status does entitle you to better room upgrades and early check-in (both upon availability), 72-hour guaranteed room availability, and a total of 20% bonus points on stays.