Outlet testers only go so far. They can produce false assurance.
One of the things that people (well, idiots -- but idiots are also people) discover when replacing an old 2-prong outlet with a new 3-prong outlet from the big box store is that they've only got 2 wires to work with. There is no ground conductor is present.
So they do the wrong thing the wrong way, and connect the ground screw on the new outlet to the neutral wire. This satisfies them ("all of the outlet parts are wired up!"), and sometimes they even think about it hard enough to justify it as being Good Enough ("ground and neutral are connected together back at the panel anyway, so it doesn't matter!").
That's bad, mmkay? Nobody should do this. Ever. It is unsafe. But sometimes people do it anyway. It's a real problem that exists in the real world.
This problem is made worse because an outlet tester won't detect this fault -- at all.
And the badness doesn't stop there, but instead compounds: The tester doesn't just fail to detect the fault. Instead, the tester will (must) cheerfully report this condition as being perfectly cromulent and safe. That false assurance is problematic in and of itself.
So, yeah: Everyone should have an outlet tester. But everyone should also be aware that they aren't idiot-proof -- their results can be poisoned by idiots from the past.
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Anyway, code. NEC 406.4(D)(2) allows replacing an ungrounded outlet with a GFCI outlet (with the ground screw disconnected and doing nothing at all). The outlet must be marked (that's why GFCI outlets include a sheet of stickers in the box that say "NO EQUIPMENT GROUND"), but it's code-compliant to do this.
So even if one pushes a landlord about an ungrounded 3-prong outlet on the wall, that doesn't mean that they're going to send someone out and tear into things to install a ground wire. It might instead mean that they put a GFCI outlet in, put a sticker on it, and call that good enough.
And, safety-wise: A GFCI used in this way actually is good enough.
But even though safe, it doesn't help at all with EMI/RFI/static issues and electronics, which the landlord doesn't have to care about. That part isn't their problem. :)
(406.4 also allows replacing existing 2-prong outlets with new 2-prong outlets, which are still being made in factories every day.)