Trenitalia have just started publishing their summer timetable. In addition, it seems ÖBB use a different timetable/database for their general timetable website than for their booking website, and Trenitalia's summer timetable hasn't made it yet into the latter. So indeed it's too early, but don't worry. These trains don't have pass holder quota and they don't fill up months in advance.

I now see that Italiarail offers bookings to pass holders for Trenitalia via a dedicated search and book page.
https://partners.italiarail.com/default?force_pass=true&aff=EURAIL&fbclid=IwAR1zG__8EtpSCsza3WI-eFahHfXJqmDhBLIZUyoO5qYeAc8t7fzjgBbTd5A

It is already showing availability for your date.

But I thought I couldn't make reservations directly with Trenitalia. Is that a new thing? 

That's Italiarail, not Trenitalia.

Although they recently wrote to me they are no longer selling pass reservations (after I mailed them about some errors in their timetables to Sicily), booking reservations with them still seems to work. They have a booking fee of €1 per order.

So is it better to just wait for ÖBB? Is there any other difference than the 1 euro fee? Can I perhaps choose specific seats with both of them? 

Neither allow you to choose seats. Although it may be possible to use the Trenitalia website ("Manage your tickets”) with a booking via Italiarail. Italiarail also want your address and a pass cover number, which is on the cover of a paper pass or which you can generate if you have a mobile pass:

When I try to book at Italiarail, it says invalid pass cover number.

We have paper pass and I use the pass cover number written on the back of the pass starting with Ixxxxxxxx

Why isn't working? 

That is certainly the correct format. Capital I followed by 8 numbers. Any ideas out there?

A few days ago, they wrote to me they don't offer pass reservations anymore. Their booking page still seems to work (we don't know for how long) but it could be the case that haven't updated the range of valid pass cover numbers. so you'd have to invent another number that works. I11111111 works. The pass cover number will probably be printed on the reservation (if you book using the Trenitalia system) but it's unlikely ro be a problem.