Train tickets for June

Your post needs a bit of explaining. https://www.trainline.eu/ is a legitimate site, and AFAIK wouldn't sell tickets for trains that are not yet available. Italy releases tickets later than most other countries, but I would expect many tickets for June to be available by now.
This is in contrast to Raileurope, for which I have heard reports of them "selling" tickets for trains before they are released, and then buying the tickets when they become available.

What "government site" are you referring to? I wouldn't expect a government site to sell train tickets, only a rail company website. And what were these tickets for, for where to where on what date? Do your tickets specify a train number, coach and seat number?

Tickets on the fast Frecce trains have been available on Trenitalia for sometime---through July when I last checked. It is the regional trains that aren't posted til later.

I have the receipts on PDF from Trenitalia. For regional and long distance trains. I paid 36US for two from Varenna to Venice. They are non refundable. But at that price I don't care. These are for 27 June. Bolzano to Varenna was 75US for two on 23 June. Happy travelers. And yes we have train, coach and seat numbers.

To tell the truth italy's Trenitalia is one of the few companies that sell trains tickets 120 days in advance, not 90 or 60, and that shows the complete schedules up to the next timetable change. Earlier than most other countries.

The complete summer schedules for subsidized trains (a.k.a. Regionale and Intercity) are always published on the 2nd sunday in June; Trenitalia has never been a minute late, for many trains they have alway been weeks ahead of the date agreed with regional goverments (those that pay the bill).

If Trainline is selling tickets that are not yet available on trenitalia.com they are following raileurope example of selling tickets that doesn't actually exist. They have never done it before and it would be quite strange from such a reputable (and authorized) re-seller.
Last night Trenitalia has uploaded another block of subsidized trains on the system, that's all.

I don't believe that either vendor is pre-selling tickets for trains that Trenitalia has not released for sale. That sounds so ten years ago, when a lot fewer trains required reservations. Rail Europe and (I'd feel sure) TrainLine both are selling through a direct database connection with most of the western European railways that they represent, including Trenitalia and Italo.

What Laura said....

It's also my understanding that the resellers are accessing the database of the rail networks that they've contracted with. If the rail network website isn't selling tickets for a particular train, then it's highly unlikely that tickets for that train will be available on the resellers website.