But that 'convenience sometimes just doesn't give you what you want!
I have an Iphone and an iPad. I have a laptop and use MS office and fire fox. I too use google flights as a search engine as it's quicker than OTA and Airline websites and gives me options I may not get looking direct at the airline. on my laptop I usually then click through direct to the airline and book direct with the airline as they have my details in my airline account (easier to deal with them direct if there is a flight change than via a travel agent if I'm not happy with the change). I then use the apps on my ipad to manage my flights.
BTW I just did a dummy booking on google and clicked the 'book with google' option and it said this
You're booking with Virgin Atlantic. Google will securely pass your traveller and payment details to Virgin Atlantic. Once your booking has been confirmed, Virgin Atlantic will handle customer support
Personally I would just book direct with VS as they already have my details via my FC account. YMMV