I would like to add my 2 cents to this question, which already has many great answers. This is rather long, sorry about that!
Discovering results is hard. It's why most discoveries are labeled theorems. It requires experimentation, trial and error, perseverance over long stretches of time. Your proposed examples attempt to address some of the aspects of research in a sandbox environment: you want students to learn the ropes of discovery but with well established, routine proofs. You hope that they will develop grit that will serve them well in their careers. However, it is likely that these will have the opposite effect: it will diminish their willingness to engage meaningfully with the material, end up with a spotty understanding of the fundamentals and ultimately unmotivated to go further in their studies.
To help explain why, I have broke down the answer in multiple parts.
Cognitive load and memory
Mathematics is cognitively demanding. It is strenuous to do mathematics at the edge of your skill level, i.e. stretching the limits of your current abilities. Working at this level implies high cognitive load, requires full attention and complete focus. When someone is learning new mathematics, they already are exercising grit. They are already discovering the results for themselves, because it is new to them and their mental schemas.
Cognitive load is the mental workload you have to deal with when learning. It is split into two categories: intrinsic and extrinsic cognitive load. Intrinsic cognitive load is the work necessary to understand the material and directly contributes to your learning. Extrinsic cognitive load is useless work that does not contribute and instead actively hinders learning. The objective is to maximize intrinsic and minimize extrinsic cognitive load. It is tempting to think that this approach will do that, but I believe it does the opposite.
Research shows that we have a very small working memory (WM), which holds information about what we are doing. Usually this means 3 to 4 unrelated items at a time. It is no surprise that mathematics easily fills one's working memory and becomes taxing to learn. There's a catch: you can bypass the limits of working memory by using knowledge and schemas stored in your long-term memory (LTM).
Someone learning new mathematics does not have knowledge and schemas stored in their LTM related to the new area. Therefore, they will struggle to incorporate important details in long-term memory by themselves, let alone think creatively and experiment to discover the established results you want them to.
Flow state and success rate
As Jason Siefken has pointed out, it's important to make a distinction between a reference and a textbook. In a textbook you want novices, with respect to the subject, to be able to confidently apply definitions, lemmas, theorems, know established examples and counter-examples, reason in terms of the new structure and understand how it fits into an overall big picture. It does not mean working at the knowledge frontier of that particular area.
With this in mind, I feel it's important to help learners achieve a flow state, a concept created by Mihaly Csikszentmihalyi where their perceived challenge is high but so is their relative skill level, in a way that they have relatively high success rate but not perfect.
Scott H. Young has a good article discussing what he calls the 85% Rule for Learning. From his article:
If you’re always successful, it’s hard to know what to improve. If you constantly fail, you won’t learn what works. Only when we have a mixture of success and failure can we draw a contrast between good and bad strategies. [...] The research also suggests that the optimal success rate for fostering student achievement appears to be about 85 percent. A success rate of 85 percent shows that students are learning the material, and it also shows that the students are challenged.
Your proposed changes will turn out to have much lower success rates I assume, drawing from both personal experience learning mathematics and as a teacher.
Thinking mathematically and problem solving
In the book Thinking Mathematically, the authors distinguish three phases into the work of dealing with a problem:
- Entry phase,
- Attack phase,
- Review phase.
The entry phase is for answering questions like "what do I know", "what do I want" and "what can I introduce". As Kostya_I mentioned, these are in line with Bloom's taxonomy first two tiers (remember, understand). This is where the learner should be occupied understanding what is being asked, what are the nuances of the problem, etc.
The attack phase is when you start to conjecture possible ways to solve your problem, together with justifying and convincing yourself that the steps you followed are correct beyond doubt. This is where you construct examples, specialize cases, think of lemmas, try to generalize from concrete forms, etc. This phase should correspond to the next two tiers in Bloom's taxonomy (apply, analyze).
Last but not least, the review phase is where you check your resolution, reflect on key ideas and key moments, and try to extend these to wider contexts. These are the last two tiers in Bloom's taxonomy (evaluate, create), where you can create new problems, solidify understandings, build bridges between concepts.
My point in elaborating all of this is that this is difficult enough when you have clear start and endpoints, let alone when you have no idea what should be of interest and what's not. Naturally, research begins like that: you have muddled understanding of something and you want to make headway in a challenging environment. But this is not the point when you simply want students to learn and internalize well-known mathematics that should be foundational.
I also recommend Alan Schoenfeld's 1985 book, Mathematical Problem Solving. Amongst other things, he discusses how knowledge impacts problem solving by analyzing verbal data together with sketches from novices, semi-novices, more experienced problem solvers and experts at fairly elementary tasks. He tracks time spent in six episodes, or stages, labeled read, analyze, explore, plan, implement, verify. The data is very interesting and shows how differently they handle uncertainty in problem solving.
I believe the best way I've seen to introduce this sort of "result discovery for themselves" is done in both volumes of Multidimensional Real Analysis by authors J. J. Duistermaat and J. A. C. Kolk. They tend to break rather long problems into many subparts (I think I've seen up to twelve items!), each involving different concepts and fitting into a sort of chain of exercises throughout the two-volume set. They serve to both apply concepts, develop interesting results, and get your hands dirty in meaningful ways. This is one example:
I think you can more easily switch to this type of problems, which will enrich your problem sets and your students' abilities. As a bonus, they get to broaden their mathematical horizons!
John Mason, Leone Burton, Kaye Stacey - Thinking Mathematically
Alan Schoenfeld - Mathematical Problem Solving
J. J. Duistermaat, J. A. C. Kolk - Multidimensional Real Analysis, Volume 1 and Volume 2
Scott H. Young - The 85% Rule for Learning
Jan Plass, Roxana Moreno, et. al - Cognitive Load Theory (somewhat outdated)
John Sweller, Jeroen J. G. van Merriënboer and Fred Paas - Cognitive Architecture and Instructional Design: 20 Years Later - This article is the most recent I know regarding cognitive load theory from a broad perspective
Mihaly Csikszentmihalyi - Flow: The Psychology of Optimal Experience