I too was very annoyed with this app's behaviour, none of solutions worked for me so I came up with my own.
In fact, Logic checks only a few files from all "essentials" (17 in my case), and they are listed in /Applications/Logic Pro X.app/Contents/Resources/logicproXXXX.plist where XXXX is the version of app, in my case it was 1023. It's listed under "Packages" key, you can find it by searching for a directory name you're concerning of, such as "Ultrabeat Samples". It also has a "IsMandatory" flag. So I came up with list of files that Logic checks on startup and if any of them doesn't exist or has unusual size - it shows the annoying download window. Here is the list of files for my version of app, they're only 34 megs total size:
/Library/Application Support/Audio/Apple Loops/Apple/01 Hip Hop/Let's Roll Beat.caf
/Library/Application Support/Audio/Apple Loops/Apple/02 Electro House/Dark Clap Topper 2.caf
/Library/Application Support/Audio/Apple Loops/Apple/03 Dubstep/Yeah Yeah Topper Beat.caf
/Library/Application Support/Audio/Apple Loops/Apple/04 Modern RnB/Zip Line Bass 02.caf
/Library/Application Support/Audio/Apple Loops/Apple/07 Chillwave/VHS Sunset Synth.caf
/Library/Application Support/Audio/Apple Loops/Apple/08 Indie Disco/Wistful Disco Guitar.caf
/Library/Application Support/Audio/Apple Loops/Apple/09 Disco Funk/Wah Power Guitar.caf
/Library/Application Support/Audio/Apple Loops/Apple/10 Vintage Breaks/Wet Disco Break.caf
/Library/Application Support/Audio/Apple Loops/Apple/11 Blues Garage/Funky Blues Guitar 02.caf
/Library/Application Support/Audio/Apple Loops/Apple/Apple Loops for GarageBand/80s Dance Bass Synth 06.caf
/Library/Application Support/Audio/Apple Loops/Apple/Apple Loops for GarageBand/Acoustic Slide 02.caf
/Library/Application Support/Logic/EXS Factory Samples/Drum Kit Designer Consolidated/Drum Kit Designer Claps/AP v1 Claps_consolidated.caf
/Library/Application Support/Logic/EXS Factory Samples/Drum Kit Designer Consolidated/Drum Kit Designer Kick/Stereo/SoCal Kit/SoCal Kit Kick2_consolidated.caf
/Library/Application Support/Logic/EXS Factory Samples/Drum Kit Designer Consolidated/Drum Kit Designer Shaker/AP v2 Shaker_consolidated.caf
/Library/Application Support/Logic/EXS Factory Samples/Drum Kit Designer Consolidated/Drum Kit Designer Tambourine/AP v2 Tambourine_consolidated.caf
/Library/Application Support/Logic/Ultrabeat Samples/Drum Machine Designer/After Party/Block_AfterParty.aif
/Library/Application Support/Logic/Ultrabeat Samples/Drum Machine Designer/Trap Door GB/Stomp_TrapDoor_GB.aif
And yes, you can completely delete folders "Drum Kit Designer Cowbell" and "Drum Kit Designer Snare" from "Drum Kit Designer Consolidated". The other directories has a file to check by Logic, except for "Apple Loops for GarageBand": it has two files.