Do we really need another Tendo Ice Bridge? Let alone with the second one being a common?

The intention is to reprint Tendo Ice Bridge, but to give it generic flavor. In light of recent Vivid lands (and generally better duals than during the kamigawa era), I feel that Tendo Ice Bridge is a card that could easily be a common. The card also is forced to slot in at common unless I want to add another common land (possible) or a common artifact (not done in core sets).

The reasoning for including this card is that I'm designing this as a core set to use with another block I create. There was a need for a larger pool of dual lands, particularly in non-allied colored decks.

You've noted that common artifacts aren't done in core sets. The only common in any core set since at least 7th that uses counters is Primal Cocoon, and then it only uses a generic +1/+1 counter. Even as you move up in rarity, the only counters you see before Rare in core sets are +1/+1. Only Phylactery Lich and Djinn of Wishes exist in core sets with counters other then +1/+1.

I think you are being too restrictive by saying what hasn't been done before. It is more important to ask yourself "why hasn't this been done before?".

I feel counters used in this way (and +1/+1 counters for bloodthirst) are very grokkable, thus are acceptable at common. The counter usage on this card does not contribute to board complexity. It is always easy to figure out what state this card is at, and the flavor idea that it has some abundance of mana energy that you can strip away for one-time-gain is pretty strong.

Artifacts are left out of common in core sets to preserve their mystique. This is clearly not the case with counters.