This has been frequently requested in the past, a better way to influence when buildings upgrade. Right now, it only happens if you level up a city, or if you unlock a tech that in turn unlocks an improvement. Cities also have a set of unique tags to upgrade when a certain amount of population has been reached, you could try using those for upgrading (but I doubt it will work).

OK, just tried it exactly how you wrote it.  Once the Beacon of Hope was finished, the tech appeared in my Completed Techs window, the research techs available down the line appeared on the Warfare track, and plate armor appeared in the shop.  There was no popup notification about this, however, so under the current situation, it happens invisibly.

As for removing the techs from the tech tree, but still having them appear, that's an experiment for another time.  Anyone?  Bueller...

This gives me an idea for building on Shards automatically giving you the appropriate element's spell list (or perhaps a tech that makes Fire Spells available for research), but that's something for later.

For now, I'd suggest using buildings, as that's been proven to work.  Some structures are tied to city level, so if you tie your tech to a structure that becomes available at a certain city level, that'll sorta kinda accomplish what you are trying to do.  You could create new buildings that only take 1 turn to complete, but those would take up space you might need later.

Perhaps if you had an upgrade to the Beacon of Hope at L2, etc. to unlock your other techs?

As for the 'invisible' tech tree, this is certainly possible.  Simply name a new category (StructureTechs) for the new techs.  Unless called out by the interface, this category will remain invisible.  Well that's how I plan to try it anyways...