No, absolutely not. SX OS is illegal in both the US and all of the EU. It violates the DMCA in the US and the EUCD in all of the EU. Even if you are using backups of games you own, although in this case the damages awarded to nintendo would be negligible.
Jailbreaking consoles in the US is illegal, you have no right to perform those modifications. Multiple court cases have been brought to over turn that decision but the courts have up held that it is illegal. They have decided that jailbreaking phones is fine but jailbreaking consoles is not.
If you bring a case against nintendo on the basis of being banned due to using SX OS then you are exposing yourself to court action.
Nintendo successfully sued a lot of companies over DS flash cards, the defendants raised the possible legitimate use of them for running homebrew and it was rejected. Customs were seizing the cards and then passing the details onto nintendo. I assume the switch dongles will need to become bigger business before customs pick up on them. In the meantime nintendo can safely ban anyone they discover is using them.
First sale doctrine doesn't apply to digital licenses, instead you are entering into a contract that involves you agreeing not to hack the system. A court is unlikely to force Nintendo to honour a contract in your favour that you have already broken.
Or they counter sue for DMCA violations and get their high costs paid by you because of your unreasonable and vexatious claim. You would need to disclose enough details for them to investigate why you were banned, they can turn up in court with a list of what you did.
I wouldn't risk it & why would you bother anyway? If you're already pirating then you don't need eshop.
If you actually read the text of the DMCA, it is vastly overblown in terms of what it actually covers. It requires two elements to violate the law. First, you have to defeat some kind of encryption or copy protection, second you have to distribute it for commercial gain/profit. Sure, if the entertainment industry wanted to buy sufficient legislators to change an "and" to an "or" it would make most of this moot. However, they haven't done that... as yet.
You can modify the hardware all you want once you physically own the console. The US FTC recently sent a nasty letter to a bunch of companies, including Nintendo, saying how their "warranty void if removed" stickers were illegal. People have a legal right to repair their own electronics or make modifications to it. There's nothing Nintendo can do about it legally. And at least with how I understand the RCM hack to work, there's no breaking of any encryption, it's a means of bypassing the encryption completely. The US Supreme Court decision involving Sony, I forget the actual case name, means that as long as there are at least some legal uses of a device, it cannot be considered illegal. Of course these days, the legal system in the US has become politicized like everything else, and people are appointed to judgeships based on the political ideology, not respect for the law, and all the things people generally agree they want in a judge.
Also, contracts are only valid so long as the terms of the contract are legal, as has been pointed out in this thread I believe. We could sign a contract where you agree to be sold into slavery by me if I buy you a Nintendo Switch. No court is going to uphold that contract in the US or EU where slavery is illegal. It's getting a bit too far off-topic, but there are a number of ways you could attack those click-wrap licenses.
I am also talking about small claims court for a couple hundred dollars. The costs of countersuing would again probably be more than it would cost if they just didn't show up in the first place. It'd probably cost them $2-$3k (roughly £1.5-£2.5k) just to pay a lawyer to draft a complaint, go down to the courthouse, file it, and then serve you; all for what might be case for £350. Maybe if hundreds of people all started doing it they might find it worth the investment, but the first few people to do it likely would win by default. I would be very interested if someone with the means and the inclination were to actually put some of the larger issues to the test. It's not me, and with a GameFly subscription there's little need for me to pirate anything. If I did, hypothetically, I would use it as a "try before you buy" sort of arrangement. If I liked the game, I would go out and buy it some time when I saw it on sale. I also think that if Nintendo had a service like Microsoft's All Access, where every single Switch game was available for a flat monthly fee, they would practically print money. If Netflix should have taught the business world anything, it's that people are willing to pay a reasonable fee if they get convenient access to quality content.
BTW, wanted to say I appreciate someone who can present a reasoned argument. All too often by this point, someone has been compared to Hitler, and the debate, as it were, has devolved into little more than petty insults. It's nice to be able to have an actual discussion with someone.