No links to research, but too long for a comment. This is an alternative approach, which doesn't have the word "proof" in it, but will create proofs as part of it. Proofs can sometimes (not always) be just a symbolic game, but abstraction is motivated, tested and has wide application.
tldr; they are already proving things every day, we need to teach the process of abstraction rather than just proof.
Proofs that will be accessible to your students are just the algebra version of abstraction. Abstraction of patterns is applicable across high school curricula.
First you need to recognise the proofs that your students do in every lesson. Every line of algebra deduction is a proof that the original expression or equation implies the new expression. In geometry, when they draw 2 identical circles which intersect, they have proven that those points of intersection are equidistant from the centres of those particular circles.
I'm guessing what you are looking for are proofs of more general results. You are looking for abstraction. Once you acknowledge the proofs they are doing you can teach how to generalise them.
For instance, students can "prove" several special cases of a more general proof that you want to encourage, using simple algebraic deduction.
$$(x+2)(x-2) = x^2 + 2x - 2x -4 = x^2 -4$$
$$(x+3)(x-3) = x^2 + 3x - 3x -9 = x^2 -9$$
They have proved 2 things already! But it looks like there is a pattern in their proofs. Maybe they can make a more general proof? They can introduce a variable where a constant was.
$$(x+n)(x-n) = x^2 + nx - nx - n^2 = x^2 - n^2$$
This equation would be the start and end for a traditional proof, but using abstraction it is just the mid-point. They have now proven several fairly concrete equations and one more general equation in a way that is motivated and easily accessible.
We now need to test the applicability of the abstraction and maybe extend it. For instance, at this point they probably see this only as a proof for speeding up expanding brackets (students often see equations as implications or rewritings rather than bidirectional equivalence). They can abstract this further by flipping the equation around to prove that certain quadratics can be factorised.
$$x^2 - n^2 = (x+n)(x-n)$$
They can test it for $n^2 = 25, 49$. Irrational numbers are extra for experts, $n^2 = 10$. See how far they can take the general proof.
They have now proven how to factorise a monic quadratic of a certain form. As $x$ is so often used as simply an independent variable of an expression, they probably don't realise that it is generalisable further. So next they can experiment by replacing $x$ by another expression.
They now have:
$$(u+v)(u-v) = u^2 -v^2$$
After all of this they have completed possibly pages of proofs, some very specific, some more general. They have further investigated the meaning of the proof to actually understand how it can be used.
I personally don't see much point in writing down simple proofs, but the process of abstraction and application of these abstractions is key in mathematics, algorithms, and across most school subjects. The above described process will definitely deepen understanding and lead to higher order thinking.