Looks like I figured it out why iPad is so unpredictably stubborn sometimes. And I have found a real solution (not such pseudo-solutions as "turn off autobright" or "autofill").
The reason of getting 169.x.x.x ip is that router's DHCP-server is too slow for the jumpy iPad, and the server cannot give the iPad a correct ip (something like 192.168.0.x) quickly. So the jumpy iPad makes decision to get itself a 169.x.x.x.
The reason why router can't manage to do it is: it's busy with other tasks.
And that is why the full router-reset helps for a while. When the router's RAM gets busy with junk tasks results in DHCP low performance, so it's too slow for iPad's DHCP client. And that's why getting new modern AirPort Extreme helps too.
So, the real solution is: speed-up the DHCP-server on your good old router (installing latest firmware), OR just set up another DHCP-server in your subnet.
For example, I have a NAS (DNS-323), and i just turned on its DHCP-server and the 169-problem was just gone. It (DHCP-server) runs so fast, even for the jumpy iPad.
The subnet config is: good old router D-Link DI-642 192.168.0.1 (with DHCP-server's range: 192.168.0.100-199), NAS D-Link DNS-323 192.168.0.100 (with DHCP-server's range: 192.168.0.200-254).
So, now iPad just gets its ip from the fastest (second) DHCP server: 192.168.0.200.
You can set up second DHCP-server on your macBook or somewhere/somewhat else.
I think your missing the point in all of this.
Yes, while it appears you must go through a small mission to get Safari to go online, I would hardly call it a "real solution".
For, what are you going to do when you are in friends/family's house? Or in an office or store? Your going to ask permission to tweak their router's settings? I think not.
Why not just add some other HD iPAD web browsers like I mentioned above that would require no new setting changes (less headaches).
Further, why is that with these other browsers, old or new routers seem to work and yet Safari can't get online. I think you should consider the idea that Safari is NOT the only browser that can get you online on the iPAD.