I've got top status in several programs, and have for sufficient years to have formed an opinion about the program and hotels' desirability. Let me give you some gut reaction to each.

If I read your original post well enough, you can qualify for status based on affordable stays near US airports. You want to spend your points at aspirational properties internationally. My comments keep that stay pattern in mind.

1. Hyatt - I like their base Hyatt Place brand. Also, some full service domestic Hyatts are surprisingly affordable. I've been a diamond for several years now, and have marched thru many of their aspirational properties. I love the Park Hyatt brand and have had stunning stays at many of them. There are other great properties; I like some of the Andaz properties such as the one in Los Angeles and the NYC Wall Street one. I love my Hyatt treatment, from free breakfast, free internet, confirmable suite upgrades, suites that can be confirmed on award stays for 60% additional points (3 night minimum stay). Lounge access is guaranteed and the international ones are very nice. Park Hyatts have no lounges. Hyatt has a limited portfolio; there's no way around it. I'm running out of new ones to build trips around. It can't be a solo program; there just aren't enough of them. But as one of two, it is awesome.
   
2. Hilton
   I've had diamond status for at least 10 years now. I've been letting it lapse the past 3 years or so, but keep getting it back due to coupling of promos/fast tracks. Their properties can be found everywhere thanks to their multiple brands. Hilton has done a good job of penetrating smaller towns in the US with the Hampton and HGI brands. That makes it easier to maintain status. I've had some great stays over the years. Conrad is an excellent brand, and I've most enjoyed Asian properties. Overall, I'm bored with Hilton properties, I don't like NOR1 and what it does to upgrades, and I don't like the current games with the diamond force and the ridiculous number of points wanted for redemptions. In my opinion, hotel quality is a notch below Marriott in the US. Lounges are mediocre in the US, and Hilton hasn't done a good job of maintaining brand standards in some countries. Yet if Hilton launches another "stay 4 times get 1 free night" I'll be right back there.
3. Priority Club/Intercontinental
   Royal Ambassador status is tough to get. There are lots of Holiday Inn Express properties these days, an excellent brand for well priced stays. It's easy to make platinum status, but that's irrelevant when it comes to the Intercontinental side. Criteria is not published. There are a small number of Intercontinental properties. They generally are what I consider to be aspirational. RA benefits include 2 level upgrades (which can be to a suite if you book the proper room level) and free mini bar. It does not include lounge access. Some hotels will give it; more often hotels charge for it. I was at the IC in Hong Kong recently. There, they charge over $100 per day for lounge access for the first person and another $30 or so for the second. You are not guaranteed the same benefits on award stays as you get for paid. I find this frustrating. Breakfast is not a benefit, unless you have club access.
4. Starwood
   Starwood has more hotels than Hyatt and less than Hilton or Priority Club (but vastly more than Interconti). It has more affordable brands in Element and aLoft; I confess I have never stayed at either. If I only had two top status chains, I would make them Starwood and Hyatt. I've been treated extremely well in Asia and well in Europe, both better than in the US. Breakfast is not a standard amenity. There are many aspirational properties around the world. I find them to have far more interesting properties than Marriott.

Given that you are interested in aspirational redemptions, you should have a look at what Hyatt and Starwood in particular offer in countries where you want to travel. If you want to go to Greece or Italy, you need to choose Starwood. If you want China, Hong Kong or Japan, there are great Hyatt properties. Thailand has a lot of great Starwood choices; Hyatt only has one property. Those are only examples to start you thinking.

I am only aware of Hyatt offering lifetime status of those I've commented on above. It requires a healthy spend over a number of years.