LICENSE.md declares that if I have an intended use in an open source application, I can apply for and automatically be granted a pure GPL licence to then copy BrowserBox Pro directly from that site. Even if only some but not all people are similarly eligible, it still becomes true that "BrowserBox Pro goes open-source". That some people are ineligible for a licence to directly copy from that site doesn't change that.
> It restricts me from using that software under the terms of GPLv3 without making my system open-source.
If your system is closed source, then you are one of the people ineligible for receiving a GPL licence from that site that will allow you to copy the program directly from that site. It is not that you get some restricted licence. You get no licence at all to copy from that site. (We'll put the non-commercial and commercial licences aside for the moment.)
Next, let's look at how the situation isn't too bad even for the ineligible sods. If by "it" you meant "your licence" then that's false as you don't have a licence at all. If by "it", you meant "that site", then that's somewhat true. The site doesn't restrict you from using the software, it only prevents you from getting a licence and downloading the software from that site. If a GPL licencee gives you a copy of his copy of the software, keepamovin automatically and unconditionally grants you a pure GPL. That is, when the copying takes that path, he will grant you a GPL even if you don't have an open-source application of your own. You can use that copy of the software in your closed-source system (to whatever extent the pure GPL's terms allow).
Now consider Alice who makes a promise P1 to keepamovin saying she intends to use BrowserBox Pro in an open-source project. This makes her eligible for a pure GPL licence to directly copy BrowserBox Pro from the site above. So she does. What prevents her from subsequently making her project closed source and using BrowserBox Pro in it? It's not the GPL but her promise P1. If she closes her own sources she wouldn't have violated her (GPL) licence but she would have broken her eligibility promise P1 and thus defrauded keepamovin into granting her the GPL in the first place. Her situation is complicated not because she has a complicated hybrid licence but because she has two legal documents to satisfy, P1 and the pure GPL. P1 was used to obtain the pure GPL. (This is where I remind myself that this is all just my point of view.)
What about Bob who has an open-source and a closed-source project? Call them O and C respectively. He promises (P2) keepamovin that he is creating an open source application, so gets a pure GPL licence directly from the site above, as well as a copy of BrowserBox Pro. He uses this in O. Can he make a copy of his own copy of BrowserBox Pro for use in C, or download a fresh copy from the site above for use in C? I believe he can do both. LICENSE.md reads "If you are creating an open source application under a license compatible with the GNU GPL license v3, you may use BrowserBox Pro under the terms of the GPLv3." Thus Bob, having got a GPL legitimately by virtue of O, can also use BrowserBox Pro anywhere else his licence (pure GPL) permits him to, such as possibly C. Using BrowserBox Pro in C will not have made his promise P2 false. So unlike Alice, he will not have defrauded keepamovin in regards to his application for a GPL licence. The pure GPL Bob received from keepamovin tells him "You may make, run and propagate covered works that you do not convey, without conditions so long as your license otherwise remains in force."
It's easy for Alice to become like Bob by having a token open source project making her eligible for a GPL licence directly from the site above.
What about Carol, who wants to use BrowserBox Pro commercially but doesn't want to pay? She can ask any GPL licencee, such as Bob above, for a copy of the program. keepamovin will implicitly and automatically grant Carol a pure GPL licence, since that's what Bob's licence assures Bob. Carol can use this copy of BrowserBox Pro commercially to the extent allowed by the pure GPL. LICENSE.md is irrelevant to Carol except for telling her that keepamovin is the author and copyright holder and the one granting her the GPL. And she needs to remember that she got her copy of the program from Bob who said he had a GPL and thus permission from keepamovin to make a copy for her.
Nobody gets a BrowserBox Pro licence from the site above that is a restricted GPL. Depending on what conditions they meet and what they ask for they either get a pure GPL or a non-commercial licence or a commercial licence or no licence at all. People who make copies of the program from GPL licencees automatically get a pure GPL licence from keepamovin. There is no restricted GPL licence in existence here.
At any rate, to repeat paragraph one, the woes of the ineligible (such as potentially you, Alice, Bob, Carol) in obtaining a GPL licence and a copy of the program directly from the site above do not make "BrowserBox Pro goes open-source" false. That at least some people are eligible for a GPL directly via the site above makes the headline true.
The GPL is pre-web so has a bug allowing web programmers to use a GPL'ed program in their web service without having to give their users the web service's source code. The AGPL fixes that bug so that such web programmers will need to give their users the web service's source code. Therefore the AGPL constrains licensees more and frees their users more.
From a quick glance, that lawsuit seems qualitatively different. It's about a hybrid licence that I'll call AGPLCCL here. One side thought it therefore wasn't an internally consistent licence and the court ruled that it was.
keepamovin is offering to grant me an unadulterated (i.e. non-hybrid) GPL licence to use/distribute my copy of his program. I'm treating as axiomatic that the unadulterated GPL is an internally consistent and open-source licence.
keepamovin doesn't have or need a licence at all (GPL or otherwise) so can't be violating it. He is distributing his program by virtue of being the author/copyright holder, not by virtue of owning a (GPL) licence for it. He doesn't need a distribution licence (from himself).
In LICENSE.MD, he declares that I (for example) may take a copy of his program provided I choose one of the licences he is willing to offer me. One of the choices is known as the GPL. Let's say I choose that. Now he's given me a copy of the program and granted me an unmodified, unrestricted GPL licence to use and distribute it. Let's write the terms down in a file called licence.gpl. (This is just a copy of the GPL itself.) I don't hold the program's copyright. I don't have a licence to distribute the program further under his LICENSE.md but I do now own a licence to use/distribute my copy of it as described in the licence he's granted me, licence.gpl.
My distribution licence (licence.gpl) doesn't allow me to add extra restrictions when distributing. So people I distribute to under the terms of my licence.gpl will automatically be granted a GPL licence by keepamovin (not me) to distribute further.
keepamovin as the author and perpetual sole copyright holder is different from me and people I distribute his program to, who are all mere licencees never copyright holders. We all hold licences he has granted to us directly and individually. (We don't grant licences to each other, having no right to do so. The GPL allows me to distribute my copy of the program to you at which point keepamovin will automatically give you a licence. (A licence is more abstract than a licence file.))
All distributions rooted in my copy should probably make clear that they are under licence.gpl not LICENSES.md. But that's something for any program offered under a choice of licences to solve.
The copy of the program at the site above is offered by keepamovin under an open source licence (GPL) and also other licences. The site thus does have an open source offering of the program. So it's true that "BrowserBox Pro goes open-source".
(Just as a licence is more abstract than a licence file, a program here is more abstract than a copy of a program. A person's giving a program to another person means ownership has transferred. So I was careful to write "copy of the program" above, as none of this distribution/redistribution involves transferring ownership of the program, which abstract work remains forever only keepamovin's.)
In case this helps somewhat, Firefox has a CLI triggered GUI for profile management and switching, which will let you run Firefox in multiple profiles concurrently: