I am in exactly the sme situation as described in this question. The solution found by that asker was
Full access !== Owner. I need to read the documentation better.
Yes. I noticed that I can only give full access to the service account. I read the doc over and over again, but I cannot find a way to make my service account an owner. So maybe read along with me? (source):
Follow the recommended steps to verify ownership of your property.
After your property has been verified, open Search Console.
Click your verified property.
Select Verification details from the Settings gear next to your verified property.
Under Verified owners, click Add an owner.
Add your service account email address as an owner to the property.
Ad 1: I verified my personal ownership of the property in question long ago (both by HTML file and by DNS record, see below).
Ad 2/3: In the search console, I can "click" the desired property from the list of all my verified properties.
Ad 4: This step turns out to be difficult. There is no settings gear next to my verified property. The closest match is a settings gear in the vertical <nav> bar on the left and about a screen length below the property selection. There is nothing to "select" from the gear, though - it is just a plain old link to the settings page https://search.google.com/search-console/settings?resource_id=sc-domain%3AXXXXX
Whatever, on this settings page, I see the following information: A section "Property settings" with three items:
There is also a button "Remove property", Next comes an "About" section about the indexing crawler. By the way, under the link next to "Users and privileges", I was able to add my service account as a user with unrestricted access - alas, as we learn, this is not enough; the only thing I can do with this user is to revoke its rights again.
The only remaining possible action that can be considered remotely related to step 4 of the documentation quoted above, is to follow the link next to "Verification of ownership". Unfortunately, that is a dead end: It lists details of my ownership verification (used verification methods, possible additional verification methods).
Ad 5: In all I tried for point 4, there is no option to add a new (delegated) owner, nowhere at all.
The lack of ability to add new owner might be explicable if I were merely a delegated owner, not a verified owner, right? However, the page shows that I am a verified owner, verified by both HTML file and DNS record.
Q: Who can help me read the documentation better and grant enough privileges to my service account to allow it using the indexing API?
Here's an amended How-To:
Follow the recommended steps to verify ownership of your property.
After your property has been verified, open Search Console.
Click your verified property.
Select Verification details from the Settings gear next to your verified property.
More precisely, in the navigation on the left, click "Settings".
4.a Click "Users and Privileges"
4.b In the user list, you should see yourself as owner; ignore any other entries. In that (=your) entry, click the Kebab menu and select "Administer Property Owners". This opens in a new Tab
Under Verified owners, click Add an owner.
Add your service account email address as an owner to the property.