1. API limits are usually per user (OAuth token, API token), not per API consumer. 5000 per user is a lot. The exceptions are ancient APIs, actually want to constrain usage, or didn't think about how their API would be consumed.
2. Honor cache headers and, headers or not, know the data well enough to cache thoughtfully.
We consume 40+ APIs for https://tacoapp.com/ and even with a full day as #1 on ProductHunt, the only APIs we had to worry about were those tied to the consuming app instead of to the user (API token) or user+consuming app (OAuth token).
Basically, if you build an API, base limits on the number of users a consuming app is acting on behalf.
If-None-Match / If-Modified-Since and using the E-Tag header.
If the resources have not been modified, or doesn't match that revision it won't count towards your limit.