The 1.7 pounds (800 grams) you get isn't going to be much bang for the money spent. You might get a better shifting groupset than whatever is currently on it. But that's about it.
Why do you think it'll be stiffer? Are you thinking in the BB because you will be going from cup, cone and spindle to something like a Hollowtech II that puts the bearing further out? I'm not good enough nor powerful enough to feel that. Are you?
And if you don't have the frame spaced to 130 mm, I think you are going to have to do some trial and error to make a 11 speed hub and cassette fit in there without the stays rubbing the chain or cog. So any wheel bought for that will be a big loss whether a budget friendly wheel or a light wheel. Frames can be cold set and 126 to 130 isn't much. Might even have enough spring in the stays, but that's still a risk if you buy all the stuff and find out it won't work.
You can probably buy a better new bike for not much more money. It might not have the 105 group on it. But in many other ways it may be a better bike.
But if you are sentimental about the bike and money isn't an issue... You can at least have a bike that will shift with the best of them today, though you will eventually find out that constantly going up small hills for longs distances the much lighter new bike will have been a better use of money. At least that's what I found out from my experience.