The original question mentions partial transfers which is a symptom others have mentioned. If you explore the contents of the device when connected to iTunes it will typically show some greyed out tracks accompanied by dotted circles. These would appear to be stalled transfers that presumably are supposed to complete on the next sync attempt, however iTunes doesn't seem to manage this. Sometimes selectively removing such items from the sync list, syncing, then adding back again will resolve, but not always. The steps I gave above should work more reliably.
If you want non-purchased music to show on your iPhone without directly syncing to iTunes then you need either iTunes Match or Apple Music. Both have iCloud Music Library where any music added to your iTunes Library becomes available to stream or download to your device over wi-fi, or cellular if you can afford the data, in the same way as your purchase history works now.
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