using a batter spreader for crepes

The Parisian street vendors use the wood spreader for A.) show and B.) because it's hard to tilt and swirl the large griddle.

Kenji covers crepe making at Serious Eats:

https://www.seriouseats.com/basic-crepes-batter-recipe

Basically you either just pour the crepe batter freehand (it's really thin) and have a few funny looking ones until you get the batter amount right or you can use the pour-back method.

One interesting comment in the Serious Eats article notes that the physics of making the perfect crepe were identified by a fluid dynamicist.

https://www.sciencenews.org/article/computer-model-physics-cooking-perfect-crepes

Just watch a video of someone who knows what they're doing (Jacques Pepin would be my first choice). After a few funny looking attempts a dexterous cook should be able to have pretty consistent aesthetically pleasing crepes.

They will all taste good regardless.

I've had and enjoyed using a cast iron crepe pan for many years.
Also you can use it for other preparations without harming its ability to make crepes.
Never had a spreader, nor needed it. If the batter is the right thinnness, you soon easily learn to swirl it in the pan until it catches the heat and forms a pancake.
The street vendors in Paris use the spreaders. Which, I suspect, does a few things for them that home cooks don't necessarily have to worry about - controls the amount of batter, speeds up the cooking, makes the large crepe (theirs are larger than most make at home), adds to the show.
So, in the end - your choice. If you like the idea of a spreader and think you'll have use for it, buy one.
Either way, enjoy your crepes.