Вы должны поместить свой тег <dx-form>
внутрь <form>
и переместить (submit)="formSubmit()"
в <form>
тег
<form (submit)="formSubmit()">
<dx-form>
<dxi-item dataField="name">
<dxi-validation-rule type="async" message="name already exists"
[validationCallback]="alreadyExists">
</dxi-validation-rule>
</dxi-item>
<dxi-item>
<dx-button text="submit" [useSubmitBehavior]="true"></dx-button>
<dx-button text="close" (click)="closeButtonClick()"></dx-button>
</dxi-item>
</dx-form>
</form>