Zermatt in early June

It's in more than two weeks: we cannot tell.
I wish you luck!

While I understand that weather is subjective, I would like to understand how is the weather generally around this time.

Weather in the Alps is not predictable more than one or two days in advance. And, even then, the forecasts can be unreliable.
June is a transition month with rain and cold temperatures still possible at the beginning of June.
You could have beautiful sunny days or it could be cloudy and rainy.
No one can tell you for sure.
There are Swiss forecasts that are considered more reliable than others.
After living in the Sierra Nevada for some years, I learned to take mountain weather forecasts with a grain of salt.

What could be my backup plan in case of a rainy / cloudy day?

Looking at the Zermatt website, it seems that the Gornergracht railway only starts on 25 June, along with several other attractions.
You may find other things to do here:
https://zermatt.swiss/en/plan-book/experiences

https://www.gornergrat.ch/en/pages/timetable-gornergrat-bahn

This says that the Gornergrat train is open. Can anyone please confirm?

Yes @Meghana Shah, the link you provided is correct. It is open. As it says:

Current Timetable until June 13, 2025 During spring season, we offer
regular connections between Zermatt and the Gornergrat. The first
ascent departs from Zermatt at 7 am, and the last descent from
Gornergrat leaves at 6:37 pm. During the day, trains run every 35
minutes, allowing you to plan your trip flexibly.

The only thing that could change this is if the weather is severe. You will also see all connections on the SBB app. Enter Zermatt - Gornergrat.

The tourist information is right at the train station in Zermatt. You can stop by there when you arrive and they can tell you all the relevant information for that exact day - what is open and what is not. They can give you suggestions for both you and for your family members with little kids.

Look here for weather
https://www.meteoswiss.admin.ch/local-forecasts/zermatt/3920.html#forecast-tab=detail-view

Scroll down and after the 8 day forecast you will see "hazards". Here is where it will tell you if there is extreme weather coming, or avalanche warning, or nothing at all. Right now there are no hazards.

Keep scrolling down to see what the average temperature, precipitation and hours of sunshine are for each month of the year.

You can do this for any location, even some mountain peaks, just change the location to the one you want to know about.

Yes, the Gornergratbahn operates year round. www.sbb.ch/en confirms this.
But it will take you 2 hours to get from Saas Grund to the Gornergrat station, so check the webcams and get an early start.
The cable at Sass Grund will be open while you are there. Worth it as a back up plan.

Saas Grund is not really a good alternative if you can't find accomodation in Zermatt. It is a two hours trip, as you are in a different valley.

Look at one of the places along the Zermatt - Visp line if you cannot stay in Tasch.

Regarding the weather: I always say that we have four seasons. And each morning the weather gods decide what season today is going to be :-)

Agreed, Saas Grund is not an alternative for visiting Zermatt at 2 hours of commute, including a bus to get out of that valley (it's not serviced by train). If Zermatt is your #1 priority, you need to up your budget to stay a reasonable distance from there to make it worthwhile, otherwise you're going to be kicking yourself for having to commute that much, especially on the very crowded buses out of Saas Grund. If your budget can't change, then stay in Saas Grund and enjoy the lovely Saas Fee and go up to the Kreuzbodensee on the cable car right from Saas Grund. It's incredible up there and if you're staying in Saas Grund, it's free to go up there with the visitor's card, which should help with your budget. All transportation in the Saas Valley, including the cable cars, is free with legal stay - a very unique situation in Switzerland.

I would caution you that, again, Alpine weather is extremely fickle. Early June weather is hard to say. It's a gamble. Make sure your hotel booking is refundable. If the forecast says bad weather, then have a Plan B.

Isnt it an hours drive?

You can't drive to Zermatt. Only as far as Täsch, where you park and take a train for the remaining 15 min. It is time-consuming!