Is there some kind of circuit arrangement that will allow you to tune to a specific frequency the same as you would with a parallel LC but by injecting the same frequency into the circuit?
That describes a direct conversion receiver. (Some might also say it describes a Regenerative receiver as well.)
That is to say, to tune to and receive 1MHz I poke a 1 meg wave into it. What I want to play with is using my function generator to select a particular frequency in the AM broadcast band and also generate a local oscillator signal exactly 455kHz above. The idea being that the two will track perfectly across the tuning range. The idea is not to listen to anything in particular but just to see how well the idea works.
Well, this is a horse of a different feather. This is a classic superhet organization where the product after the first mixer is at 455kHz and then passes through a passive filter network. It is a great organizational scheme and was the basis for most of the AM broadcast receivers from about 1940 until, probably, the 1980s or later.
But then we got into all kinds of other things. This may not answer your question, but I have a feeling it might give you some more tools to navigate through the fascinating engineering tradeoffs and technical fashion show that is radio design.
Bottom line takeaway here is:
1. There are many different receiver architectures, and it is worth giving each of them a look. There's nuggets of wisdom in technical archaeology.
The ones that come to mind: Coherer, Tuned-Radio-Frequency, Regenerative, Heterodyne or super-heterodyne, and direct conversion. In particular, there is much to be learned from the various exotic super-het schemes that used multiple levels of conversion to take advantage of one kind of filter widget or another.
2. The goal of the front-end in a receiver is to deliver energy near the signal of interest from the antenna to the some element that will eventually get us to the demodulator/detector. As you read about RX designs, pay attention to the flow of the desired signal and the interfering or undesired signal through the path. Much of what motivates good design is as much about the stuff you don't hear in the speaker.
Ideally, the front end would deliver only the signal of interest, but narrow filters bring with them a host of problems. (That's why you don't see many TRF receiver implementations (even historically) above one MHz or so.) This part of the story is about adapting engineering styles and decisions to contemporary technology, supply, and economics.
3. Almost any radio component with gain will have some non-linearity. Mixers also have non-linear behavior -- that's what makes them mixers. As Mr. C01l has pointed out, the mixer will produce lots of products that beat your incoming signals (and its neighbors) against each other and against the carrier oscillator (we're assuming a heterodyne architecture at this point). Well designed receivers take a great deal of care in balancing the gain before and after the mixer, the selectivity before the mixer, and the local oscillator injection at the mixer. (Superhets are organized to put most of the spurious products well outside the passband of the next stage of amplification.) The good news is that many of the spurious products are related to the cube of some fraction which makes them shrink in amplitude relative to the signal of interest.
This last part is, perhaps, the hardest thing for people to internalize. Nonlinearity is our friend. It does very interesting things. Try some pencil and paper experiments. Get yourself a list of trigonometric identities and start mixing some A*sin(f1*t) and B*sin(f2*t) and C*sin(f3*t) by squaring the sum of two or three of them. Then cube the sum of two or three. Take a look at what the trigonometry tells you. What happens when A is much bigger than B or C, but when B is bigger than C? How about A much bigger than B and C, but B and C are equal? The drill will take an hour or so, and grind through a pencil, but it will help embed things like no simulation or youtube video could. Do this experiment, and you'll understand the discussion of IP3 that you'll eventually get to on the path to enlightenment.
4. Since electronic components like diodes, varactors, FETs, and BJTs have non-linear responses to their inputs, it is often a good idea to put some filtering between the antenna and the first one of these that the signal sees. That means that the preselector, for most sizes of budget, should be passive, otherwise it contains a diode, varacter, FET, or BJT and so you need to put a preselector in front of it. which should be passive. <rinse-lather-repeat>
Preselectors are a great idea. Heterodyne mixers are also a great idea. But neither is a good substitute for the other.
Because of all these tradeoffs, you'll see receiver organizations that appear to be quite odd. Some have ton's of loss in the front end, having spent lots of incoming energy in filters before the first active element. This is common in HF where receiver overload is a real problem.
But then there are receivers that stick a transistor to the very first non-metalic element in the signal chain -- right out at the feed of the antenna. These often look at hot dots in a cold sky. When there are no other signals to beat with yours, preselectors aren't all that important. (But watch out for spurious signals from the microwave oven...)
This isn't a direct answer to your question, but I hope it can guide you in your search for other questions to come.