You have totally misunderstood how bonds work on a "present value" or "mark to market" basis.
(All of the below assumes the bonds actually pay as agreed. Actual default risk is something totally different, and still present here.)
When you buy a bond and hold it to maturity, you're sort of right. If you put in $10,000 into buying a coupon bond, you will get the coupons plus the $10,000 back at the end. And if you buy a zero-coupon bond for whatever amount, which will be worth $10,000 at maturity, you'll get the $10,000 back at the end.
In fact, you don't even need to "assum[e] yields stay positive." When you buy individual issues and hold to maturity, you don't really care what everyone else's yields do; you get what you contracted for.
The problem comes if you want to actually sell out of your position, OR to know the true value of your position (essentially equivalent operations) along the way.
If you put in $10,000 into a bond yielding 5% coupon, and the next day yields spike to 10%, nobody will want to buy your bond for $10,000 any more. You most certainly have lost value. "Aha," the naif says, "but I could always hold to maturity and get my principal back!" Sorry. Do the thought experiment where instead of buying the 5% issue on day 1, you instead buy the 10% issue on day 2. Compare the cumulative sum you receive under each scenario. Investing on day 1 (at 5%) is strictly worse than investing on day 2 (at 10%).
Likewise, if yields instead crash from 5% to 1% on day 2, your position will be worth much more. Your $10,000 notional bond yielding 5% will net a buyer so much more than $10,000 spent on a 1% yielding issue that she will pay more than $10,000 for it. You have had a real gain, even if not realized.
The same thing applies to bond funds or indices but with much more smoothing across a portfolio. With bond funds, however, there is not even the illusory "X dollars out" guarantee; since they are marked to market every day you might well never enjoy a breakeven price.