Hey KTM, welcome!
First of all, many new parts are crap these days. Our fault mostly, but that's another story. With these things in particular, it seems that the less you pay, the less quality you get is a true statement.
Second though, you may not have a starter or a relay (solenoid) issue in the first place.
To answer your question though, yes, a bad starter can take out a starter relay too.
But, when you're not getting power to anything, the starter system has nothing to do with it. It sounds plausible that you're fighting a wiring issue with your fusible link wire there at the solenoid/relay and maybe just messing around with it will "fix" it temporarily, only to give you grief again.
However, that's only if you have no power at to anything at all. If you have power to any of the accessories, the fusible link is not your problem after all.
The fusible link is a short section of Black wire (may have a red stripe) right there at the starter relay. Since yours is a '70, it's possible that you only have two wires on the left side large post. That would be the battery cable to the positive side of the battery and the Black wire that feeds power to all accessories, and charges the battery when the alternator is putting out.
Unfortunately, all of this is way too common after buying an Early Bronco these days. Lots of people off-loading their own problems on to a new owner. Chances are that the PO (previous owner) either didn't know about it, or knew and just couldn't figure it out.
What is the condition of the battery cables? They can get old and fail internally too, even though they look half-way decent on the outside. Sometimes it's just cheap insurance to change them out at the same time you're either changing the starter or trying to track down an intermittent fault like you're having.
Good luck. Any chance you can post up pics of the engine compartment in that battery corner? You can use one of the free online hosting sites, or post directly to the forum if you pay the 13 bucks to become a contributor.
No pressure to do it right now of course, as you just got here and might not feel you should invest in an unknown. But if you end up wanting to post a lot of images for us to check out (we love pics in case you didn't notice!) it might be worth it.
Paul