Hi Robert, thanks for your reply. You're right that I haven't looked into libsyn for quite a long time, and I apologize if I pooled it in with those other services without enough current knowledge. If libsyn is working well and has support, great! I will definitely not advise against it, and may even revisit as a possibility.
As for self hosting, I have been hosting a podcast on its own domain since October 2006 with no problems. Not sure what defines "popular", so specifically: last month 250 GB downloads, 2.6 million hits to web site, and 40,000-50,000 individual feed subscribers in any given 24-hour period. Since episode 1, these stats have fluctuated between 100-1000 GB downloads per month, 1-4 million hits per month, and 15,000-50,000 feed subscribers. I'm guessing this level of popularity is above average, but below some of the specific names you mentioned. Nothing has ever brought down my site — thankfully I don't even know what "bring down your own site" means!
A few times I have considered moving it, and did once move it to Feedburner, which was horrible (the extra stats were nice, but not essential). I do share some of the bandwidth and subscriber load with Soundcloud (adding maybe another 7,500 subscribers to the above stats.), but my web host has been great. The only issue was once they moved my site to a different server and I had trouble updating. I switched my FTP settings to the new server, and everything was fine. That happened once, in ten years. For these reasons, and for the control that I have over my own feed, and knowing that my feed will never become obsoleted or lose support, I stand by hosting my own feed and suggest that it's a great option for podcasters.