you can always get out of the sandbox. But whether or not you will rank is a different story.
I got out of the sandbox with every site that I had, and I didn't do anything to get out of it. Sandbox is just a period of time during which - for various reasons (niche too competitive and / or your domain is fresh) - google doesn't let you rank even though you are indexed. But once that period of time passes (which can be anywhere from 3 months to 24 months from my experience) the sandbox ends and you are eligible for ranking.
It's not that complicated, I don't know why you guys are so confused by this
Anyway, to address some of your questions...
I launched a good-quality website 35 days ago. But even after all this time, it still doesn’t show up on Google — not even when I search for my brand name, which has zero competition.
Even typing the full URL doesn't bring it up. It only appears if I search with quotes like "domainname.com".
this doesn't mean that you are sandboxed.
Actually, are you ranked / getting clicks or impressions for any other keyword for this website? If you do, you're NOT sandboxed. If you don't, it's 50-50 (either you're sandboxed, or the niche is really tough and you've not yet met google's requirements to break into top 100 even...
As for branded keywords, these are tricky because google will not rank you even for your own brand if the brand doesn't get enough mentions.
There was someone on the forum who managed to rank a brand new site (also brand new domain) for a completely new brand some time ago, but I forgot who he was... or what the link is. But if I find the link I'll post it here, but that was the only case that I've heard of when someone managed to rank a fresh site for a fresh brand quickly (and with black hat stuff if I remember correctly).
But otherwise you need people to search for your brand 1000s of times, over time (weeks usually) before google starts ranking you for your branded keyword, even if that keyword has no competition.
So, either find a way to have multiple people (from different IPs, geolocations, devices, etc.) search for your brand in google or via other mediums (social media or forums for example), and have them land on your site and after a few weeks of daily searches and / or mentions of your brand google should notice the activity and rank you at least on page 1 (if not top 3) for your branded keyword.
Either that, or wait for google to discover your brand on their own, although this could take months if your site / brand is not popular and doesn't get much activity
On other search engines (like Bing and Yandex), the site is already ranking at the top.
well yeah, but other search engines are not as competitive and restrictive as google
In the past, my websites would usually start ranking for their own names within a week.
so it's a recent update then.... Also, I'm thinking that your website - being so fresh - is lacking trust, and trust is a big deal with google nowadays. This would justify why in the past you managed to rank fast while nowadays you don't (trust had gained weight in the last year or so from what I've noticed, so it's very possible that unless you create authority and trust for your website it will not rank for its branded keywords as fast / easily as it used to)