Your work is copyright as soon as you write it. Registering your copyright entitles you to seek additional damages if you ever take someone to court for violating your copyright. In some jurisdictions you need to have registered your copyright in order to get into the courthouse because no possibility of additional damages means a copyright case involves such piddly sums lawyers and courts aren't interested in it.
The WGA operates a depository service. This is not connected to the courts in any way and does not have legal standing. But the WGA is a respected body, and if they make an official statement that you filed Document X with them on Date D and it has not been accessed since, that carries weight.
There is no legal requirement to display on any submission whether or not you have registered your copyright or used a depository service. Failing to display a copyright notice does not weaken your protection.
Some people consider displaying such information the mark of an amateur. Some of those will be prejudiced against such submitters as probably being difficult to work with.
Registering your copyright in no way interferes with your ability to submit your work.
Using a respected depository service in no way interferes with your ability to submit your work.
Failing to note on your submission whether or not you have taken either of the above actions in no way interferes with your ability to submit your work.
(Also note that neither registering your copyright nor using a depository service is necessary to submit your work. They offer additional layers of protection in case of dispute. But your work is copyright as soon as you write it. As the writer, you will have much better evidence you wrote it (in the form of notes, earlier drafts, etc) than anyone who tries to misappropriate it.)
If you are concerned about protecting your work, in addition to registering your copyright and/or using the WGA depository you should research anyone you are submitting to. The WGA, WGC, and most other guilds maintain blacklists of agents and producers who do not abide by industry codes of conduct. If you submit directly to producers/studios, you should submit only to producers/studios who have signed onto the appropriate guild's model or standard contract. This is not to say all producers who do not sign the model contract are shady, but most legitimate producers who do not sign do not sign because they do no business with anyone outside their business group. There wouldn't be any point submitting to them anyway.
Remember: Ideas cannot be copyrighted, only execution. I believe the WGA does not accept anything less than a treatment for deposit.