API already supports this, but it can be used only when the project that you created is a subproject and it inherits the members, otherwise, you cannot used it because the project has no members and the default assignee needs to be an existing member.
On my instance (latest trunk), the following request:

created the project with default assignee and version:
{
"project": {
"id": 12,
"name": "abcde",
"identifier": "abcde",
"description": null,
"homepage": "",
"parent": {
"id": 1,
"name": "eCookbook"
},
"status": 1,
"is_public": true,
"inherit_members": true,
"default_version": {
"id": 7,
"name": "Systemwide visible version"
},
"default_assignee": {
"id": 3,
"name": "Dave Lopper"
},
"custom_fields": [
{
"id": 3,
"name": "Development status",
"value": ""
}
],
"created_on": "2021-02-24T18:31:09Z",
"updated_on": "2021-02-24T18:31:09Z"
}
}
I'm closing this, please reopen if something is missing.