I've been traveling in China fairly extensively for the past 10 years and I now know maybe 5 words in Mandarin - in other words, most of China is very doable on your own as long as you speak English. I had the same concerns as your wife when I first traveled to China nearly 10 years ago but after deciding to do independent travel, I happily found it to be much easier than I thought it would be. English signage was much more prevalent than I thought it would be and in the most visited tourist areas (Beijing, Shanghai, Hong Kong, Xi'an, Guilin/Yangshuo, Lijiang, Dali, Kunming, Pingyao, to name a few), a fair amount of English is spoken, especially in the hotels that cater to international travelers.
It does take a little up front planning similar to what is described above for Japan, however there are many travel sites which are excellent sources of information (start off by Googling China attractions, or Beijing attractions, Xi'an attractions, etc. and go from there). When in China you do need to have the names of places you want to visit written out in Chinese. Hotel staff will do this for you and the Chinese names of most sites/attractions can also be printed out from many China travel websites before you leave home. When traveling about, you always want to have the name and phone number of hotels you stay at, in Chinese, with you. Be aware that hotel names in Chinese are almost never a direct translation from the English name so you can't use Google translator for this. Over the years I have had several taxi drivers in China call my hotel to get directions when needed -even when they didn't speak any English, so having the phone number of the hotel with me has been invaluable.
In some places it may make sense to arrange going with a day tour or to arrange a private car/driver and tour guide if needed and these can easily be arranged over the internet before you leave home or after you've arrived at a destination, so you needn't be tied to a strict schedule for your whole stay in China. I've found the planning for a trip like this to be a big part of the enjoyment.