I think you are misinterpreting what I said. Yes, you need MilesSAAver availability the whole way. Availability is usually better on short-hop flights, so you have to connect a bunch of these in a row.
Let me explain how I went through the process. I first find the trans-atlantic flight below:
Wed Jul 15
HG 8600 6:55 AM VIE 8:15 AM ZRH
AA 65 10:00 AM ZRH 1:10 PM JFK
So I want to find something leaving JFK in a generally westernly direction on Wednesday afternoon (or Thursday morning if need be) First I tried JFK to SFO, and of course no MilesSAAver availability. Try AA hubs from west to east... still nothing to LAX or PHX. But there is a flight to DFW or ORD:
Wed Jul 15
AA 4242 2:52 PM JFK 5:13 PM IND
AA 4299 7:23 PM IND 7:40 PM ORD
or
Wed Jul 15
AA 4242 2:52 PM JFK 5:13 PM IND
AA 1382 7:35 PM IND 8:50 PM DFW
Next step would be to search DFW/ORD to SFO, which comes up blank. So we search to LAX (could also search to PHX):
Thu Jul 16
AA 1312 9:10 AM ORD 10:58 AM PHX
AA 5749 12:00 PM PHX 1:25 PM LAX
Great, now to find LAX to SFO:
Thu Jul 16
AA 5833 3:45 PM LAX 4:56 PM SJC
or
Thu Jul 16
AA 218 6:35 PM LAX 8:09 PM SFO
Boom, we did it. Your final itinerary might look something this:
Wed Jul 15
HG 8600 6:55 AM VIE 8:15 AM ZRH
AA 65 10:00 AM ZRH 1:10 PM JFK
AA 4242 2:52 PM JFK 5:13 PM IND
AA 4299 7:23 PM IND 7:40 PM ORD
Thu Jul 16
AA 1312 9:10 AM ORD 10:58 AM PHX
AA 5749 12:00 PM PHX 1:25 PM LAX
AA 218 6:35 PM LAX 8:09 PM SFO
This whole thing should be 30k miles. Now my point here is that this is possible; however, unless you are some kind of airplane enthusiast, you are probably not excited about taking 7 flights in a row. You may decide that you're better off booking VIE to JFK with AA miles, and purchasing a ticket the rest of the way. However, if you do book this crazy itinerary, the good news is that if something more direct opens up, you can call and change for free. For example, if a non-stop were to open between JFK and LAX, you could switch to that and go down from 7 flights to only 4.