That would make sense, but it still doesn't seem to work.
I'm only referencing stuff from ADCore, but adding that module as a dependency doesn't change anything. Here's what I have now:
<ArcGIS defaultAssembly="ProAppModule3.dll" defaultNamespace="ProAppModule3" xmlns="http://schemas.esri.com/DADF/Registry" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://schemas.esri.com/DADF/Registry file:///C:/Program%20Files/ArcGIS/Pro/bin/ArcGIS.Desktop.Framework.xsd">
<AddInInfo id="{f2d61d01-095a-4b9a-9bf3-4437643fe624}" version="1.0" desktopVersion="2.5.22081">
<Name>ProAppModule3</Name>
<Description>ProAppModule3 description</Description>
<Image>Images\AddinDesktop32.png</Image>
<Author>NLBVRE</Author>
<Company>Acme</Company>
<Date>3/16/2021 10:34:42 AM, 2021</Date>
<Subject>Framework</Subject>
<!-- Note subject can be one or more of these topics:
Content, Framework, Editing, Geodatabase, Geometry, Geoprocessing, Layouts, Map Authoring, Map Exploration -->
</AddInInfo>
<dependencies>
<dependency name="ADCore.daml" />
</dependencies>
<modules>
<insertModule id="ProAppModule3_Module" className="Module1" autoLoad="false" caption="Module1">
<groups>
<group id="UpdateInsertTab_Group1" caption="Group 1" appearsOnAddInTab="false">
<button refID="ProAppModule3_Button1" size="large" />
</group>
</groups>
<controls>
<!-- add your controls here -->
<button id="ProAppModule3_Button1" caption="Button 1" className="Button1" loadOnClick="true" smallImage="pack://application:,,,/ArcGIS.Desktop.Resources;component/Images/GenericButtonBlue16.png" largeImage="pack://application:,,,/ArcGIS.Desktop.Resources;component/Images/GenericButtonBlue32.png">
<tooltip heading="Tooltip Heading">Tooltip text<disabledText /></tooltip>
</button>
</controls>
</insertModule>
<updateModule refID="esri_core_module">
<tabs>
<updateTab refID="esri_core_insertTab">
<insertGroup refID="UpdateInsertTab_Group1" placeWith="esri_project_styles" insert="before"/>
</updateTab>
</tabs>
</updateModule>
</modules>
</ArcGIS>