We are on Sitecore 9.2 with SXA 1.9. Installed Sitecore Publishing Service 4.0.1.
Everything is working fine but when we try to publish the content from Experience Editor, it still shows the default publishing window rathe than the one comes with Publishing Service.
After a bit of digging we figured out the root cause.
By default, Publishing Service adds “/sitecore/system/Aliases/Applications/Publish” item in the “core” database.
This alias has a “Linked item” field that leads to the “/sitecore/client/Applications/Publishing/PublishDialog” item.
If the “Publish” item is either not present or renamed, the vanilla publishing dialog is shown in Experience Editor.
When we checked into Core database this item was not present. We created the item manually and BOOM. We could see the Publshing service window in Experience editor now.
For more details please have a look at this article which actually talks about reverting to default publishing after installing Sitecore Publishing Service.
Hope this help!