Thanks for the file, Kotten_3.
If I get your target correctly, just animate the size of the Sphere to gain the desired.
Currently the Hair material is set in a way that it is tighter on the “roots” then on the tips.
Perhaps test out “Sketch ‘n’ Toons”, as it has the options to set parameters based on many parameters.
Screenshot, Thickness:
https://www.amazon.com/clouddrive/share/BTOVOo21QEptsIOgOiE6mpoxOXdPu85sT6X3qtLFixs?ref_=cd_ph_share_link_copy
With those options in mind, one could use a Gradient set to 3D, and the starting point would be then the camera. The radius would be then in need to fit the desired distance. In this way you can fine-tune the Thickness (as well as other channels in the Hair material) in any way you like.
File:
https://www.amazon.com/clouddrive/share/P0Sr7awdBfKBHErIACefZvFWyHZHxFf5D23jMp8eC0?ref_=cd_ph_share_link_copy
Screenshot:
https://www.amazon.com/clouddrive/share/y6dg9hFhai4e0y6tinkZ5p7tuzQFCmWKvWg8Mu4laVq?ref_=cd_ph_share_link_copy
It doesn’t show the camera view, but the camera in relation to the spline/hair combination. Switch to camera to get a final view to this.
Please always the core problem in the file, not all the extra stuff, thank you.
BTW, to share 3rd party products in your file public-ally is perhaps not a good idea.
All the best
P.S.: yes long renders, but not bad for a quick sketch:
https://www.amazon.com/clouddrive/share/Ri21su47N965R6UdCfDO5gHzuiKNlnL8i7iwcaZ1Mwm?ref_=cd_ph_share_link_copy