I doubt Tile was providing any sort of meaningful competition that would spur Apple on anyways.
My suspicion is that Apple started designing Airtags due to their own internal research / feedback showing the numerous problems tile hardware (and software) was causing for their users. However, Apple isn't doing this to muscle in on Tile's turf just because.
Rather, what Apple is doing is lay the groundwork for eventually digitizing our non-smart devices by making them visible in location layers. AR applications will be able to utilize these location layers since they will serve as a foundation for adding context to our surroundings. In this context, it doesn't matter which brand of tracker you use so long as it feeds into the "find my" network. Either way, Apple wins.
Once that is established, release AR glasses and the wearables market is pretty much in the bag for Apple.
What Tile will likely end up doing (assuming it stays in business) will be to continue appealing to android users (pretty much the Pebble Playbook). We will then see a similar Apple Watch vs Pebble / Fitbit dynamic play out, where airtags become the default choice for iOS users, and Tile for Android users, with neither crossing paths ever again because they are each effectively targeting completely different markets.
But because tile trackers are effectively an end in themselves, rather than a means to an end (ie: AR glasses), this may also be as far as they go. Tile will never have the resources to build their own "find my" network the way Apple has. My guess is that they eventually get acquired by either Google or Amazon eventually, and this antitrust lawsuit from them gets tossed out the window.