Right now the ESRI CDN source looks like it has a broken build. For instance, you cannot pan or zoom on the map right now whereas yesterday you could. I would wait until later in the day to see if their fix doesn't take care of your problem.
This probably has nothing to do with Leaflet — it requests tile images the same way as other JS libraries, and caching is fully handled by the browser. Check your browser/devtools settings.
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