Should 4 year old NK be able to write her name?

At this age, she’s almost 4 and a half, how much writing practice do you do with your NK? Shes in preschool and a couple of her friends work on their letter work book (where you trace the letter with dry erase) often but their nanny is strict in making sure they finish their letters, and as a result they are able to free hand most letters. We’ve done our workbook with them occasionally and we talk about how her friends are able to do their letters because they practice a lot, but I don’t make her do it, it was more she wanted to do it to join her friends and then would lose interest after a page.

Occasionally, I suggest doing it and either she says she doesn’t want to or she will do a letter and then draw all over the page. I’ve never forced it too much (but rather focused on the more we practice then we will be able to write our own name) because I definitely don’t want for it to be associated with any resistance.

Should I be making it a part of a daily routine like “once we do our letters then we can do this”? Should I just keep it casual? She still needs correction with her pencil grip and hasn’t gotten the hang of keeping her hand on the page. Should I maybe focus on fine motor activities instead (we sometimes do scissor work which she is more enthusiastic about) and coloring? (She’s gotten a bit more precise with her drawing but it is definitely still abstract lol just scribbles)