I am allergic to bee stings, myself and, to be honest, I don't agree with you.
The best solution is the one you can actually carry with you – if you can't afford the EpiPen, this system would be far, far better than nothing.
Further, while anaphylaxis is a life threatening situation, it isn't a situation where, if recognized in a timely manner, 10 seconds more to pull out the second or even 3rd home-brew auto-injector would kill the patient.
Next, the widespread existence and feasibility of a home-brew option would itself exert significant pricing pressure on the original EpiPen thereby making it more affordable for everyone and thus less likely that I would even need to actually make one.
Lastly, I would, in fact, use the above hypothesized home-brew option for my kids or wife (if they were alergic), if we couldn't afford an EpiPen or as a way of making sure we always had one available.
As I argued, having multiple devices cheaply on hand makes failure of one an okay thing. If they have a 10% failure rate (which itself is a high failure rate even for this scenario), having 3 on hand would mean you'd have one-tenth of one percent odds of all three failing. Add a 4th and you get down to 0.01% – get the individual reliability down to 5% failure rate and 3 devices would give you 0.0125% odds of failure. Those are actually pretty good odds. Moreover, I'm not sure what the traditional EpiPen's reliability is, but I'd bet it's not that much better than a 0.000125 failure rate.
I think that rules it out. Imagine a family member, or anyone, dying because of your sub-optimal solution.