I’m struggling to figure out what is it that you actually want. You can consider induction on an arbitrary well-founded relation instead of an ordinal (and only on well-founded relations, as induction actually implies that the relation is well-founded). (This covers all the various special cases like transfinite induction, structural induction, $\in$-induction, and whatnot.) If the relation is reasonably encoded, its induction scheme should have the same proof-theoretic strength as induction on the ordinal which is the rank of the relation. Thus, the only thing you can achieve is to have ordinals represented nonuniquely by elements of a fancy, more complicated structure. As a matter of fact, this is what you do anyway, since e.g. in the usual representation of ordinals below $\varepsilon_0$ in arithmetic using Cantor normal form, ordinals are identified with certain trees. So the answer to your question appears to be “yes, just call them trees instead of ordinals”.
My view is that the large cardinal hierarchy already has all the
principal features of the project you are proposing.
Each of the large cardinal concepts can be regarded as
corresponding to a certain conception of the set-theoretic
universe, if one should entertain the von Neuman hierarchy up to
such a cardinal, and this makes a perfectly good universe. Every inaccessible cardinal $\kappa$, for
example, gives rise $V_\kappa$, a transitive model of ZFC and a
Grothendieck universe in fact. Every Mahlo cardinal $\lambda$ is a
limit of many inaccessible cardinals $\kappa\lt\lambda$, and the
models $V_\kappa\subset V_\lambda$ have much the same relation as
what you describe in your question. If $\lambda$ is Mahlo, then
the smaller models $V_\kappa$ for inaccessible $\kappa\lt\lambda$,
which are perfectly good set theoretic universes, each extend up
to $V_\lambda$, a larger universe having what it thinks is a
proper class of inaccessible cardinals (and hence also the
Universe Axiom). Indeed, when $\lambda$ is Mahlo then the
collection of inaccessible cardinals is not merely unbounded in
$\lambda$, as you request, but also forms what is known as a
stationary class in $\lambda$, meaning that it intersects
nontrivially with every closed unbounded set. This seems to extend
and refine the idea of your cofinal tallness. Similarly, every
weakly compact cardinal is a stationary limit of Mahlo cardinals,
and if $\delta$ is a measurable cardinal, then not only are the
weakly compact cardinals below $\delta$ stationary in $\delta$,
but they form a set of normal measure one, a much stronger notion.
This reflection phenomenon is nearly universal in the large
cardinal hierarchy, where properties of the larger large cardinals
reflect down to robust classes of the smaller cardinals. The
strong cardinals reflect in this way down to the measurable
cardinals, and the Mitchell order carries this idea still further.
Supercompactness reflects down to superstrongness. It is an
intensely studied phenomenon.
In this sense, the subject of large cardinal set theory is already
undertaking your project. What we are studying is precisely how all the various large cardinals can be construed as smaller universes extending into larger ones. For the large cardinals that are axiomatized in terms of the existence of certain embeddings $j:V\to M$, this extension process proceeds in two ways: $M$ is larger than $V$ in the sense that $\text{ran}(j)\subset M$, and $M$ is smaller than $V$ in the sense that $M\subset V$. It is the interplay and tension between these two sense that gives rise to much of the power of these axioms.
I would say that this includes elements of algebra, broadly
construed, if one regards the direct limits and large systems of
large cardinal embeddings that arise in the theory as having an
essentially algebraic aspect. Surely the extender embeddings
concepts developed in the theory of canonical inner models exhibit
a fundamentally algebraic character.
And the subject is hugely involved with philosophical
considerations, which guide the choice of new large cardinal
axioms as well as motivate or attempting to explain why we should
believe that they are consistent or true. One can say infinitely
more about this.
Best Answer
If by cardinal you mean an initial ordinal (an ordinal not equinumerous with any smaller ordinal), then your new scheme is merely an instance of the scheme for ordinals. Indeed, you can see it clearly as a special case, if you realize that under AC every cardinal is $\aleph_\alpha$ for some ordinal $\alpha$, and so your proposed cardinal scheme is asserting that if $P(\aleph_\beta)$ for all $\beta\lt\alpha$, then $P(\aleph_\alpha)$ holds. (And actually, the new scheme is equivalent to the old scheme, if you consider replacing $\alpha$ with $\aleph_\alpha$.)
Without AC, however, there is a more general concept of cardinal, by which is meant something like the equinumerosity class of a set. If $Y$ is a set, then a smaller cardinality amounts to a set $X$ such that $X$ injects into $Y$ but not conversely. Without AC, these cardinals are not necessarily well-founded. Thus, the transfinite induction scheme fails for these more general types of cardinals.
For example, it is consistent with ZF that there are infinite sets that are Dedekind finite, so that they are not bijective with any proper subset of themselves. Let $P(X)$ assert that $X$ is not infinite Dedekind finite. If $Y$ is any set and is infinite Dedekind finite, then for any $a\in Y$ the set $X=Y-\{a\}$ is strictly smaller in size (else there is a countable subset of $Y$ by iterating the bijection) and $X$ is also infinite Dedekind finite. In other words, the property $P(X)$ satisfies the induction scheme for cardinalities, but does not hold of all cardinalities (if there are some infinite Dedekind finite sets). So transfinite induction for cardinalities can fail without the Axiom of Choice.