> However, if *actual* infinity is a *necessary* part of TM model, then > it cannot be equivalent to lambda calculus, because there is no > infinite component in any lambda calculus expression,
A TM does not have an _infinite_ tape, it has an _unbounded tape_. This means that the tape is guaranteed to be as large as you need it or, equivalently, that the tape will grow as you need it but always stay finite.
It isn't hard to write a TM simulator in pure lambda calculus. The converse is harder, but possible.