The "current path" is not "a huge extrapolation", it what is going to happen if we continue to do what we do (it includes our currently made and actually planned measures):

https://climateactiontracker.org/global/temperatures/

The current political "climate" makes very improbable that we'll achieve "only 2 deg in 2100" target that the politicians mention but don't do enough -- compare with how much easier were had we started in 2000:

http://folk.uio.no/roberan/t/global_mitigation_curves.shtml

The optimistic and best case scenarios are also scary.

Extrapolation is when you have one wife on your wedding day and decide you will have thirty wives next month: https://xkcd.com/605/

What we have with climate is more like your neighbor appearing with two wives two months ago, with thirty wives one month ago, and today he has sixty wives and maybe a few husbands. Still waiting to verify if something is not right?

Yes, and annoyingly it only goes back 20,000 years. Going back 400,000 years would tell a much more interesting story.