After a couple of phones were lost/stolen back a few years ago, I got one of these from my Mum for Christmas and it worked perfectly - also backed up texts (I'm afraid I can't remember the make). I think though some email programmes now can sync phone numbers (which isn't quite the same as back up but does the job) - I now do this weekly from my phone to yahoo contacts.
Why not just transfer all the numbers from the sims to the phones and then put 1 sim in to each of the phones in turn and back up from the phone to the sim. You can then go back and clear all the phones. That way you dont need the sim reader.
one issue is that the sim cards usually have limited capacity. i think mine can store only 100 contacts. also none of my contacts are on the sim, but instead on the phone.
your phone might have come with software to download all contacts into outlook or some database file which you can then load into your next phone.
this is the only reason i have outlook installed and have done transfers this way many times.
I am actually trying to recover all the numbers that have been stored on all the phones over the course of the years. Nokia, Samsung, Apple, HTC - I have plenty of cross-platform incompatibility. I just want to get all the data back onto one computer and then start to sync to the iPhone and be done with the whole mess.