To get to Pullach by train you will have to go through Munich train station. While this is a large train station, with a lot of luggage it is probably not much fun to go from the long distance trains to the S-Bahn. However, the S-Bahn to Pullach runs often enough that you do not have to hurry. Just take your time.

To get to Munich is easier from FRA than from STR or ZRH. The fast trains (ICEs) to Munich all go through the ICE station in FRA, i.e., you hop on the train, find some space for your luggage, and relax until you are in Munich. I take these trains quite often and in the late mornings when the flights from the US arrive they are usually not crowded. I still suggest you get a seat reservation, though. The train station in FRA is very nice and designed with travelers in mind. There are escalators and lifts, so you will not have a problem with your luggage.

From STR you have to take the S-bahn to the main station, and you even might have to switch trains. While the connecting S-bahn will depart from the same track, with a lot of luggage this could be an issue. There is an elevator from the subterranean S-bahn station in Stuttgart Hauptbahnhof to the long distance tracks, but it might be difficult to find and is quite slow, so there are often lines there. The alternative is getting up to the main station from the S-bahn by escalator, but this is a pain, especially if you do not know your way around, since the signage is not that good. Alternatively you can take a taxi to the train station. This will take about 20 minutes or so, depending on the traffic on the Weinsteige in Stuttgart, but is probably faster and more comfortable than the S-bahn. Much more expensive, though. However, even with the S-bahn, the trip to Pullach will take less than 4 hours. In addition, I really like the airport in Stuttgart a lot. Although STR has grown a lot, it is still a small and personal airport. And the connection STR-ATL has been much better for me for op ups than FRA-JFK and FRA-CVG (even many years back when I was a GM only).

For the ZRH to Munich connection, there are direct EC connections from the airport to Munich (the 4:44 hour long connections). The train ride from Zurich to Munich is very beautiful, in my opinion, but the frequency of connections from ZRH to Munich is not that high, so if you miss your train, you might have to wait for a couple of hours. And the train ride to Munich is longer than the one from FRA. The alternative connection goes by train and ship. Very nice if you have the time, but not advisable if you have lots of luggage.

So, overall, if I were you, with a lot of luggage I would probably fly into FRA, and only if that's not possible I would fly to STR because of the shorter travel time, overall.