I expect the Garmin is drawing so low a current that the battery pack has decided its time to switch off.
Different battery packs will have different minimum thresholds and will behave differently.
This. An eTrex uses very little power, compared to the charging phone the battery packs are designed for. I measure a draw of about 70mA from my eTrex 30 with the backlight off and 110mA with the backlight at full brightness. (I did some proper measurements with a µCurrent and an oscilloscope a while ago, which I detailed in a post somewhere that I can't now find.)
Second the point that one of the main advantages of an eTrex is that you don't have to do this USB-power-on-the-move nonsense. Mini-USB sockets are neither waterproof nor durable, and AA batteries are cheap, readily available, easy to swap and give somewhere over 20 hours runtime (decent NiMH recharagable or primary Lithium cells, alkalines are disappointing).
FWIW my Anker PowerCore Speed 20000 has been powering my eTrex 30 (sitting on the menu screen with with the backlight and GPS receiver switched off) since I started writing this post, and shows no signs of switching off. It will also stay on and power my phone after it is fully charged (which I consider an advantage, YMMV) where my Aukey PB-T9 and EasyAcc PB12000A did not. Might be a bit overkill for using on the bike, audaxer-who-bought-an-Edge-style, but on the other hand it's an excellent choice for touring, as it appears to do the Right Thing in response to fluctuating input (as from a dynamo charger or solar panel) and in combination with a QC3 compatible wall-wart, it can charge at 15W.
ETA: After >10 minutes, the Anker was still powering the eTrex. I tried the Aukey and it switched off after about 30 seconds. The EasyAcc switched itself off even more quickly. That's all the power banks I have to hand.