eventually kill my battery?

Eventually, yes, but it would probably take quite a long time. When they are not charging a phone, they should draw very little power.

I have this iOttie wireless charger plugged into my 'always on' 12V outlet, and I am not worried about draining my battery. A few times I didn't drive the car for 3+ days, and the car started up just as quickly as it normally does.

I use an earlier version of this; iOttie Wireless Car Charger Easy One Touch Wireless 2 Qi Charging Dashboard Phone Mount for iPhone, Samsung Galaxy, Huawei, LG, Smartphones https://www.amazon.com/dp/B07X4YDKTF/ref=cm_sw_r_cp_api_glt_i_KV61ST2RDKNX995VW0MJ?psc=1

Mounted on my dash, below my rear-view mirror.

I use it regardless of navigation or not, as it charges my phone at the same time.

https://www.amazon.com/dp/B07X4YDKTF/ref=cm_sw_r_apan_i_B1TY9T4XM5CQR9N5MG09?_encoding=UTF8&psc=1

I've had this one for the past year and it's great. I can't say that it allows access to both ports at the same time but it has legitimate fast wireless charging and I have used wired headphones while it's wireless charging in the stand and it works flawlessly. Its on the pricier side as far as these go, but it's been worth every penny.