I have a modified C40//02, which is effectively identical to your C70. I don't think the problem you are experiencing has anything to do with characteristics of your guitar.

A Classical isn't optimal for the exercise, but I just tried it to satisfy curiosity and it's doable eliminating equipment as the cause. Don't apply too much thumb pressure. It's just not necessary with Yamaha or D'Addario normal tension nylon strings despite their height from the fretboard.

Other possibilities are either as you surmise, or the entire exercise including gradual development of flexibility and strength is new to you. When you refer to "really painful", are you experiencing moderate chronic discomfort or acute pain?

Give it a rest for a day or two if the former, and see if it goes away. Stop and seek professional medical advice for the cause if the latter.