Looks a bit low to me, but it's your house...your call.

If you are going to re-do it, the sooner, the better. The grout will only get harder with age.

You'll need to remove it to at least 2/3 the depth before regrouting. Use a new sponge, well rung out this time and try not to be as agressive with the cleaning. Looks like you might have overworked the grout lines last time...

What's the profession way to remove floor grout?

Multitool (Eg. multimaster, multimax, jobmax) with grout blade and a vacuum.

But since it's new, you could first try a handheld grout saw.