This is correct; Persistent Symbols get moved from scene to scene, and thus their DOM parentage changes. Browsers have the behavior that they will reload iframes in this case. It can be demonstrated with code like:
<button onclick = "boxB.appendChild(myiframe)">Switch</button>
<div id = "boxA" style = "border: 1px solid; width: 320px; height: 180px">
<iframe
id = "myiframe"
src = "data:text/html;charset=utf-8,<script>document.write(Math.random());</script>">
</iframe>
</div>
<div id = "boxB" style = "border: 1px solid; width: 320px; height: 180px">
</div>
A differing number means a reload has occurred.
I agree with the suggestions above that Symbols (and thus Persistent Symbols) are in general the use case for this feature. That said, organizationally I definitely understand that a full document can be better since it can have multiple scenes and can be easier to edit.