Why not, though?
They're obviously at least partially physical constructs objects given that Shiro could get an arm graft from future Shiro and the magical circuit lineups were a plot point.
Then you have the worms, crests which can be transplanted despite being bunches of magic circuits, Rin transferring her magic crest to Shirou that one time, pedobait Jack's Surgical Procedure is implied to be at least partially mundane, and spiritual surgery can manipulate them too. I'm sure there's several other examples I'm missing, as well.
On top of that, it says explicitly that Rasputin's internal furnace allows him to integrate materials he swallows into his body. That seems like a sufficient loophole to be able to jack magic circuits. You did say Fate was all about loopholes, didn't you?
I'll accept getting nothing from bird because I misunderstood how familiars work apparently but I would argue that the theory is still sound.
That's the meshing of a spiritual body and physical body, which only worked because they're the same person. Circuits are physical and in the soul simultaneously, the issue isn't that you can't swallow circuits and integrate them in your body. The issue is that after you do that the circuits are going to reject your body and freak out which will cause you to kill yourself in what's basically the soul's autoimmune response.
So maybe I shouldn't say you can't integrate circuits so much as you shouldn't.