"First, do no harm."
I have no problem with schools gently doing some academics at nursery stage, but ONLY if they actually know what they are bloody doing.
I get very cross when I hear of nursery school staff etc. "introducing literacy" when actually all they are doing is buggering things up and creating confusion in the kids.
First point. They shouldn't be sticking whiteboard pens into these kids' hands to write. If they want to encourage writing readiness then fine--but give them proper small pencils and pens, and paper, and show them how to sit comfortably at a little table. (And they can do this by drawing.) Otherwise all you are doing is teaching crap habits. Whiteboard pens are too fat to fit comfortably into most little children's tiny hands, and they encourage wrong gripping habits which real teachers will later on have to spend time and energy correcting.
Secondly, "teaching kids how to write their names" is just about the silliest activity you can possibly do if you are trying to introduce phonics. English names are notorious for odd spellings based on higher-level rules which tiny children are not ready to cope with. So unless a kid's name is Jen or Sam, all this is going to do is cause confusion. Teaching a kid to write "Sophie" and then getting her to point at it and say "Suh! Oh! Puh! Huh! Ih! Eh!" is not teaching her to read--it will only make her really confused and convince her that relationships between letters and sounds are arbitrary. This will slow down the teaching of proper reading.
Nowadays, it is usually advised to teach only lower case letters until kids are blending properly with these (because when kids are introduced to upper case ones at the same time, it encourages mixing up and confusion of upper/lower case, which teachers then have to spend yet more time sorting out). So they shouldn't be introducing upper case right now either, yet if it's "names" then I guess they are.
It's also recommended that children learn the smooth precursive style for each letter, not ball-and-stick, so that teachers don't have to waste time teaching one way and then unteaching it and teaching another way. Is this nursery teaching the kids proper smooth precursive style for each letter? (Genuine question. Perhaps they are, perhaps they are not).
Making the kids do this before they can go out and play does not sound right either. These kids are really young and the nursery should not be inviting resistance and negative attitudes to learning in this way.
If they want to do some school readiness stuff, they should do some reading up on "phonics awareness" stuff and do some enjoyable games and activities that will help children practice splitting words up into sounds. If they don't want to do this kind of genuinely helpful activity, then they really shouldn't be doing anything at all. It is fine not to do any literacy at all at nursery age!