Yes, E-3 is out of date. But it's a hell of a lot better than nothing & not counting the USAF there are 20 of them operating in Europe, & it isn't the only AEW aircraft operating in NATO or soon-to-be-NATO Europe: there are 4 Turkish E-7, 2 Italian G550 CAEW, & 2 Swedish & 4 Greek Erieye, plus 3 French navy E-2C.Getting a little off track from the Bundeswehr...
The problem with the E3's is they are all based off the same aircraft and that aircraft logistics are essentially dead. As dead as a 707 airliner.
So its not just one or two airframes that have high airframe hours, the entire fleet is fast approaching the stage where it will no longer be possible to be flight worthy. The large number of airframes still operational is actually a huge problem in depleting spares and logistics. Due to a lack of spares, items that need replacement come off any are remanufactured and then put back on, which takes time and money.
This has been an issue for a long time, the US was aquiring aircraft for spares since the 90's.
707 based aircraft were the most expensive aircraft for the USAF to operate. The E8/E4 were running around nearly $90m a year! In 2018. The only aircraft that comes close is the B2 bomber.
The E3 is also technically outdated. It is not designed for a modern 21st battlefield with stealth aircraft, low observable missiles, drones, modern jamming and spoofing. E7 has extremely good sensor fusion, and great situational awareness. It can operate in a complex battlespace with peer adversaries.
A replacement for E-3 should have been ordered years ago, for the reasons you give, but at least something's being done. Italy's getting more G550 AEW, Sweden's going to replace its current SAAB 340 Erieye with Globaleye, & perhaps increase numbers, & the UK's buying E-7 (converted from secondhand airframes to speed up delivery, in addition to plans to replace E-3 - eventually.