investigation has not found a wifi sheild that has a 'good' reputation of being bug free
I think this is correct, they are overpriced and flakey, hence my comment about using a WiFi repeater, particularly since you already have an ethernet shield. Having said that, you make no mention of the power supply, it may be an issue, and I am not aware of a repeater running off battery.
Here is what I working with right now. I wrote this program, and am now trying to get rid of the wired cable and the router. I dont need the arduino to act as a router, just let one device, access it.
The goal is far from clear but, if it is just to perform some tasks as shown at 00:29, and this is a one-to-one exercise, you can be sure you can do the whole thing in bluetooth and there is a whole raft of bluetooth apps just waiting for you at the Android apps store. There are bluetooth devices that have a range comparable with WiFi
I think the exact term for what i want to do is a peer to peer server/web browser set up, with the arduino running a wifi board and the iphone/ipad/computer/smart device/
I think "peer to peer server" is a contradiction in terms.
The other thing about the approach is that it would be app free on the other end, hence the desire to continue the use of an browser.
If you can back off from that principle, life might be easier............