I always forget exactly what is permissible with the LGPL, but isn’t it the case that the whole point is that using a LGPL library doesn’t “infect” your program?

Generally it means you can't (in this case) mess with the Qt Libraries, you use them as is, and link to them dynamically. You do NOT have to release your source code, however you do have to provide Qt source (libraries) to anyone who requests them from you to whom you "have distributed" your software to (paid or unpaid), whether you provided a download link or DVD or usb drive, etc. If you someone received your software outside of your distribution channel you can tell them to go pound sand though since you didn't distribute to them directly and they got a copy through some other means.