Yes. Is there any way in which I can reach the staff?
Just a suggestion

Bear in mind, it would only be 1 of +1000 players. But any suggestion would carry more weight coming from a chess.com admin. (hint hint)

There’s a way to make it work by creating a chess.com title of UTT Champion (UT). The hard part would be the rotating bit where it only lasts a week.

I heard back from @jdcannon. He likes the idea and will run it by the events people, and if they OK it he'll check with the dev team. That's likely to be the biggest hurdle.
If it does get approved, it would be an easy and fun award that they could use for other events too, not just Untitled Tuesday.

one more thing that needs change is the timing. They are always at horrible times for Indians
That's something that will never happen

The reason for the timing is that I don't want to overlap with Titled Tuesday. Some people like to watch Titled Tuesday and then come play Untitled Tuesday.
To change it, then, would mean moving it earlier, which would put it in the middle of the work day for the Americas. The other option would be to move it even later into the Indian morning, when it's in the middle of the night for Europe.
We have Saturday events on a rotating time schedule to cater to different time zones. When those events are at a good time for Eurasia and a bad time for North America, hardly anybody shows up. So my impression is that we just don't have very many people in Eurasia who want to play.
If those Eurasia-focused Saturday tournaments start to fill up, that will tell me that we have more people from those time zones that would play an earlier event. Until then, I don't think it makes sense to change the time to make it easier for a few people that mostly don't show up.
I think the two key sticking points would be actually getting the person into the tournament (which might require some development work) and getting the site to approve it. If it's announced clearly and made a regular thing, titled players would get used to it pretty quickly.