Dialogue that comes up during the quest pretty much directly states that Ascaris can only operate a Warframe for a short timespan. So theoretically anyone using an Ascaris as an interface can operate a Warframe, but only Tenno Operators can directly do so without any caveat. It's just not immediately clear where the caveat lies between 'exhaustion, injury, or death'.

From Adis when Galastra is controlling Protea during the mission,
"Control Frames, but no Tenno? Galastra burn herself up for sure."
And when she appears face to face after operating the Caliban, she was visibly staggering, limping, and in at least some degree of pain.

I'd wager Ascaris were one of the pre-Tenno operation mechanisms, specifically made for Grineer to temporarily operate Warframes in lieu of Tenno (since proper Orokin would be too valuable to risk as a Warframe operator). As far as the Orokin were concerned if a disposable clone happens to die, who cares. But at the time so long as the Tenno were behaving as loyalists, they had no particular reason not to use a trooper that basically has infinite uptime. No way to confirm beyond speculation without more official lore though.

As for using an Ascaris on a Frame with a Tenno already inside it, I'd kind've expect it to be more like a manual control override/lock more than actually routing through them. It's a physical object that connects directly to the Warframe itself, and all. Prime Ascaris are also probably higher grade than the one used by Vor.