The client will have to rely on their evidence that they have a contractual arrangement with the freelancer to create the thing, if they don't have a license for the product. Permission to use the freelancer's stuff is required, which could be in the form of an explicit license, or an implicit license arising from the fact that the client paid the freelancer to create the thing (which implies "and let us use it", though this would have to be sorted out in court). Suppose for example you have an exchange of emails where you say "We'd like an X", and he says "I can do that for $N", you agree, and you pay him. That is sufficient to establish his permission for you to use the thing. Signatures for such agreements are really not required: what's needed is evidence that there was an agreement in the first place.
As for DMCA takedown, freelancer files a notice and asserts the item was used without permission, then client can file a counter-notice. At that point, freelancer has to either file suit, or else the allegedly infringing material gets put back up. Client only has to prove something in court, when being sued by freelancer. Emails, for example, showing that there was an agreement would be useful – if you have no proof whatsoever that you even had such an arrangement, that would be a bit of a problem.
In general, if a person wants to make use of and distribute stuff created by another person (where the other person would hold the copyright), it is in the client's interest to protect themselves from lawsuit by getting a license to use the material. However, client probably needs to hire an attorney to draft a generic license agreement. Licenses like the CC licenses generally say "Anybody can use this, with certain conditions", which freelancer presumably would not agree to (and could be contrary to the interest of client). There are very many possible approaches: assign copyright to client; grant client a permanent exclusive license to us in any way they want; grant client a nonexclusive license to use in a very specific way for a couple of years, etc. Essentially, there is not and cannot be a one size fits all approach to restrictive licenses.