My guess is that the cost to ship it in, get it fixed (or get a refurb), and get it back is just short of the cost of buying a new Arcade. And if you have a system without HDMI, they're going to send you a refurb without HDMI and it will probably blow up again anyway.
I'd say if you're not playing it that much anyway, best to do without it. Otherwise, get a new Arcade that will have improved internals and HDMI. Of course, I might be overestimating the cost of getting it repaired.