Twitter offers several free endpoints for developers, including search, streaming, a raw sample of tweets, and the like. They all have rate limits and the like attached, but they're real, free, access to their data.
For people who want bigger, broader, deeper access, they sell access to that data. That's been the case for the last few years.
In this case, Emojitracker unfortunately falls into the cracks in between, where they need deeper access to do what they want, but it's not a commercial enterprise.
I agree, it would be nice of Twitter to continue giving Emojitracker free access, but are they obligated to do that? They are a business, and providing access to millions of Tweets for free seems like an unfair obligation to ask them to continue indefinitely.