If your goal is to use traditionally correct grammar, use "you" rather than "yourself".
More info here [0].
[0] https://www.chicagotribune.com/lifestyles/ct-xpm-2012-09-05-...
See for instance https://www.collinsdictionary.com/dictionary/english/by-your...
> by yourself
> PHRASE
> If you do something _by yourself_, you succeed in doing it without anyone helping you.
> "I didn't know if I could raise a child by myself."
My understanding is that "yourself", "myself", etc. are reflexive pronouns. I.e., they're used when and only when referring to the very same person as the sentence's subject.
The examples you mentioned fit that pattern. The message on the computer screen doesn't.
Who is being by somebody (the subject): you
Who is the subject being by (the object): yourself
The grammar in the original sentence is correct, you're just getting confused about which word is the verb. "Repairable" is not a verb, it is an adjective.
Another way to phrase the same sentence would be "by yourself, you can repair the system". Another phrasing would be "you can repair the system when you are by yourself". You would not say "by you, you can repair the system".
"Repairable by you" is a different, also grammatically correct sentence, with a different meaning. "You can repair it" and "you can repair it on your own" (aka "by yourself") mean two different things.
If you want to say "I think the phone is repairable by myself" it's grammatical, but strange because people don't usually think with assistance.
subject: I
verb: think
object: the phone is repairable
adverb: by myself
edit: I think I'm agreeing with you, but I couldn't tell what you meant by "the original sentence."
I'm certainly not in a position to argue in favor or against the question if "by yourself" could ever be "correct" grammar in the passive voice. An easier question would be whether "you can repair it yourself" sounds cleaner than "it is repairable by yourself". And there I'd certainly agree.
I/could/raise (by myself)
Instead try:
"I didn't know if a child could be raised by myself."
Child/could/be (by myself?)
It's the difference between "you could repair (by yourself)" and "phone could be (by yourself.)"
Ain't is my favorite example. It was considered proper English until the lower class started using it [0]
I'm not against formal grammar but to grammar nazi everyday language is to be nothing more than an elitist pedant who presumes form to be more correct than function (which is only true in creative writing imo)
Note what I wrote:
> If your goal is to use traditionally correct grammar, use "you" rather than "yourself".