Properties of a term rewrite rule
In general, there is no obstacle to define a rewrite system as above, and a few sources do indeed require rules 1. and 2. only as additional hypotheses.

However the hypotheses are almost always necessary (but not sufficient!) for both termination and confluence, which is indeed the main object of study of rewriting as a field.

Perhaps it's useful to remind why this is the case:

Termination:


If $x\rightarrow t$ is allowed as a rule, then any term $u$ rewrites to $t[u/x]$. In particular, the system is non-terminating.
If $l\rightarrow r$ and $r$ contains a fresh variable $y$ not in $l$, then for example, $l\rightarrow r[l/y]$, and so the system is non-terminating.


Confluence:


Take $x \rightarrow f(x)$ and $g(a)\rightarrow b$. Then
$$ g(f(a))\leftarrow g(a)\rightarrow b$$
for instance, breaking confluence. In particular, you can "insert" an $f$ around any term you like, which makes confluence very hard to satisfy.
If there are constants $a$ and $b$, then given the rule $f(x)\rightarrow g(y)$, you get
$$ g(a)\leftarrow f(x)\rightarrow g(b)$$
again, breaking confluence.




Now every theorem involving either termination or confluence could be prefaced with "assuming 1. and 2. holds, ..." but it makes sense to restrict your attention to systems where the theorems aren't trivially false. I don't think anyone would be shocked by a paper which lifts these restrictions though, at least condition 2. (condition 1. really creates anarchy). Condition 2. is mostly about non-determinism, for which rewriting is a nice framework in general.

There are some systems in which free variables can appear in the RHS, but usually not any fresh variable $y$ is allowed: there are additional constraints which need to be satisfied, e.g.

$$f(x) \rightarrow  f(y)\ \mid\ 0\leq y &lt; x $$

where $x$ and $y$ are allowed to be integers. These types of systems are called constraint rewriting systems and are extensively studied.