How can such a thing as a "fully collateralized" Bitcoin loan exist?

Without a reference the exact instrument you are considering, it is not possible to give a precise answer. there are two sorts of things you could be talking about:

Crypto Loan

If you want to buy a house for $50k and have a bitcoin, you can get a loan of US dollars and provide the bitcoin as collateral. A way this often works is to transfer the bitcoin to a 2 of 3 multisig wallet, where you, the lender and a third party each hold a key. This allows you to buy the house but keep your investment in bitcoin, so if it goes up you still get to keep the profit. The problem is you have trust the third party, if they and the lender get together they can take your bitcoin. Obviously if you default on the loan the lender gets your bitcoin legitimately.

Take a short or long position on bitcoin

Some exchanges allow you to take leveraged positions. Here you deposit you one bitcoin, and borrow either 10 bitcoin for a month and sell it for USD (short) or 10 * value of bitcoin in USD and buy 10 bitcoin (long), using the 1 as "collateral". After a month you reverse the purchase, pay back the loan and will have gained or lost money depending on the change in value of bitcoin. The problems here are that A) you have to trust the exchange and B) if the value changes too much the "wrong" way your position will be closed out and you are likely to have nothing. I looked at this a bit, and I think there is nothing else that is more like gambling that is not regulated as gambling.

What you describe is what a loan is. The collateralization scheme seems off, though.

You can't, generally, just get things for free. If you want to buy a house, which is for sale at the price of one Bitcoin, there isn't generally a way for you to get the house without spending one Bitcoin. It doesn't matter if you spread the transaction out over time, as with a loan repayment schedule, or if you transfer the entire coin at the time of purchase.

This is exactly the same as buying a house using dollars (or any other currency). If it costs $250,000 there is no way for you to get the house without expending $250,000.

Where this looks a bit odd is when you consider Bitcoins to be the unit of exchange (you are buying something priced in Bitcoin), the thing being loaned, and also the collateral for the loan. The typical collateralization for something like a loan to buy a house is the house itself-- if you default on the loan, the lender takes the house and sells it to recoup the loss on the loan. So in the example in the question you would retain your Bitcoins, but would lose the house (valued at ~1 BTC) to satisfy the loan.

If you don't default, then you have to repay what you borrowed plus any interest charged. It doesn't matter where you get the repayment money. It is exactly the same if you earn another BTC through a day job or if you rent out the house and thereby earn 1 BTC which you use to pay the loan back. Most people don't have "genius ideas" that drop windfalls of cash on them. They work and, over time, pay off the loan balance piece by piece.

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Is there something I'm fundamentally missing here?

Yes. You can't, generally, buy something without spending money (or exchanging something of similar value). I find people often become confused when talking about cryptocurrencies because they start to conflate their asset-like qualities with their currency-like qualities in ways that sometimes don't make sense. It would be an odd arrangement to borrow $100,000 in cash by putting up $100,000 in cash as collateral. That a person would prefer to get the $100,000 loan and never pay it back is understandable, but it's not realistic. Why would anyone, ever, take the other side of that deal?

What is the point of taking a loan to "not spend your Bitcoin" when you have to pay it back anyway?

This is nonsense if you consider BTC to be currency (which I do not). It is less unreasonable if you think of cryptocurrencies as assets. If you have 1 BTC which is worth $100,000 (a made-up number), and expect it to increase in dollar value (for example, you think it will be worth $110,000 in one year's time) then you might prefer to hold the coin than to spend it now: if you buy a house by selling or transferring the coin, you gain the house (valued at $100,000 or 1 BTC at the time of purchase) but lose the coin, and so you "miss out" on the extra $10,000 the coin is worth in one year.

So you might put up the coin as collateral for a loan of dollars, and then pay back the loaned dollars over time. Then you don't spend the coin and get to capture any increases in the coin's dollar value. If things work out well, you end up ahead-- the coin is worth more than what you spent to buy the house: you have 1 BTC (now worth $110,000) and one house (worth $100,000). Even after paying back the loan (with $100,000 cash or with ten elevenths of a BTC when one whole coin is worth $110,000), you own assets worth $10,000 more than what you started with.

But using "your" coin as collateral for a loan of a different coin, which you borrow and then spend, doesn't change the fact that you've spent a coin. And you owe your lender for the coin you borrowed. No one has donated a coin to you, nor given you a free house worth one coin.

Is the idea that the borrower assumes that they will make a lot of money somehow, unrelated to the loan, and be able to pay it back until it expires?

Yes. There are other types of loans, like business loans, in which the loaned money is expected to directly contribute to allowing the business to generate enough revenue to at least pay back the loan (plus interest). But the expectation that comes with loans is generally that they will be repaid, some way or other.

Is it really that "simple"?

Yes, and you don't need the quotes around "simple". Loans aren't free money for the borrower. In every case I've seen where people try not to "spend Bitcoins" they are expecting their Bitcoins to increase in value (in terms of some specific currency) and so don't want to expend those coins in exchange for things which they expect will not appreciate in value (also in terms of some specific currency) as much as they expect the coins to.

Upper_Case's answer is great, but I just want to focus on the very first part of your question to point out what I think is the misunderstanding, because it's right at the beginning:

Let's say I want to buy a house, but I don't want to spend any of my 10 BTC. So I want to borrow 1 BTC.

In order for me to not be able to scam whoever lends me that 1 BTC, I have to "fully collateralize" the loan, by locking 1 BTC into this system.

Stop right there. You are proposing to borrow something (1 BTC) by giving the exact same thing (1 BTC) in collateral. That is what does not make sense. Getting 1 BTC in exchange for 1 BTC is a complete no-op. It is like giving someone a $1 bill and receiving another $1 bill in return.

There isn't even a reason to call this a loan, because nothing is changing in the financial situation. If you have $10 and want to buy a hamburger but don't want to spend any of your $10, do you get a loan for $1 to buy the hamburger and collateralize it with one of your dollars? No. You already have $10. Getting someone else to give you one of their dollars in exchange for one of yours changes nothing. You had $10 before, and you have $10 after. You can just buy the hamburger right now without involving any "loan".

It doesn't make sense to borrow what you already have. The idea of collateral is that you give someone something you have (or the right to take it under certain circumstances), and they give you something you don't have instead. For instance, you give the pawn shop your guitar and they give you $100. You do this if you need $100 but all you have is a guitar. If you have 10 BTC and what you want is 1 BTC, you already have what you want, so there is no need for you to borrow anything.