Yes. Absolutely. Autograbbers are SO worth it.
Eventually you'll have more animals and chasing them all down every day can become a chore. Anything that can automate even a small portion of your work is worth it. The Autograbber qualifies for this. So does upgrading your barn or coop fully, to get the Autofeeder that Ereo mentioned. Once your pets love you, you can easily skip days of taking care of them in order to get other things done... but without autograbbers milking your cows and goats and sheering your sheep, you could miss production.
Buy one for every barn or coop you own. Srsly. (Don't ignore purchasing other things just to get one, but make sure to eventually get one for every one.)