Firstly, I feel your pain that so many don't seem to understand the difference between import and sync. Every time this question comes up in any forum, people always provide information on how to import, not sync. Not to mention, we're talking contacts here people, not calendars, we all know about CalDAV.
With that said, I've come across two possible solutions... sort of. First, there's a fairly decent email client - eM Client - that hooks directly into iCloud. No, really. The updates from iCloud to eM Client are quick (it appears as though eM Client will poll iCloud every time it senses the user is in the contacts section and/or the app regains focus - not sure if this is true, but it does appear this way). I have not tried eM Client to iCloud sync, and well, I'm not going to... more on that later. eM Client also has quite a nice interface, a fast universal search, smart folders, etc, etc. Oh - and two-line email sender/subject entries, unlike Thunderbird. For personal use it's free - up to two email accounts. For anything beyond that, it's $49.95. For iCloud contacts, it doesn't presently sync iCloud groups. It also works with Gmail and syncs Gmail contact groups, although predictably the respective contact pools (iCloud, Google, etc) are separate and not cross-synced.
The second solution is 'Contacts Sync for Google Gmail with Auto Sync' by Playa Apps. It can be found in the iTunes app store. I haven't tried it yet, but it seems to offer auto syncing on an iOS device that will in effect sync contacts between iCloud and Gmail, including the groups. There's also the option for one-way sync, probably a good idea as things can become quite ugly when trying to sync Gmail contacts to iCloud (or for that matter any non-Apple contacts to iCloud). In essence, one could manage their iCloud contacts within the Apple realm (iCloud website, recent Mac or iOS device), sync one way out to Gmail, and then use gContactSync to sync Gmail to Thunderbird. Voil?, iCloud contacts in Thunderbird, via Gmail. The price of the app presently is $2.99.
I myself have been trying to achieve true, non-duplicate iCloud contact sync/propagation into a non-Outlook Windows email client over the years - finding periodic, short-lived solutions (as in free or at least affordable - no, Plaxo, I'm not paying you $5 per month ad infinitum) - and now it looks as though there might be a couple of viable alternatives!
Hope this helps.