YOUR FAMILY IS GOING TO HAAAATTTTEEE IT.
Oh they won't say it to your face, they will just ask if they can keep the old computer "as a backup" and just use that once your gone. I know this. I know this from 10 years of experience of helping my grandma, her friends, her friends friends, and working at Unisys as a drive around tech for both enterprise and consumers. My american born Korean friend knows this as well and has tried hard to find similar solutions. At the end of the day, all people want to surf the web, go to links friends and family talk about, and emal/text. Computer literacy ONLY helps the safety of said activity's. When my grandma discovered her grand-kids were all on Facebook, she didn't care that she set up 3 accounts with different passwords, but I was able to strengthen that out by having chrome force sign her in one. My friends mom would constantly complain about the linux install because it wouldn't play flash properly, so he had to switch her back to XP. These are just a few examples I can mention, and you WILL get these calls.
Just remember, any "teaching" you do must be "with" them. Not "to" them. From their perspective, everything is working on their end. Their computer might be selling their information to the highest bidder and telling everyone how their penis can grow larger with one payment to the Nigerian, but they can still get pictures from family, they can still get messages. They can still see the latest football game scores.
Do the wifi idea. Hell do one better. Set up a small embedded system with a built in modem. They sell embedded boxes, but an old p4 with a modem should work. PFSense is something I would suggest. I forgot the package but there is a way to set it up to act as an email proxy. Have it drop all attachments that aren't images. It can even unpack zip files and check if its just images in those. Have it dial in daily keeping the email box clear. Just say it just checks your mail every day at 1am or such. Its like 4 am and I am half out of fuel and I am sure you have looked at some of these products. They will still get viruses as I don't think it will have the bandwidth to keep the updates up, but at-least you can have it track their surfing habitats and can block country's they have no reason to go to. Talking to you Russia and most of Africa:P
I hate to say it, but this is the best way for someone who is computer illiterate and doesn't use the computer much to care. You come in saying "just plug this between the computer and the modem to make the internet faster/safer" They don't feel alienated on the limited computer knowledge they learned and you can, they keep the computer/interface they are used too, and at the very least, have some control over the data going in and out.
Whatever you decide just ask yourself, "Does this improve their experience? If not, what would happen if they just chunk it and not tell you?"
As a side note, if it wasn't to much of an extreme luxury for the country its going to, I would recommend also an iPad for the interface. That single thing has introduced text messages, Facebook and god help me gaming to my grandma. If it wasn't for the large constant patches it needs and the very high chance it would get stolen, I would recommend it too.