The short answer that the risks are the same, or very close. Differences arise primarily because the fund manager will likely rebalance to maintain a target duration while you most likely will not.
You appear to be confusing bond risk with uncertainty about a bond's payout. The latter is only part of the former. A bond that is absolutely certain to pay out its contractual obligations can still be quite risky. That is, its price can fall because the discount rate we apply to those payments has increased, as it does when the rates on competing bonds increases.
But, you may ask, what if you hold the bond to maturity? Isn't this "risk free?" No. When the bond's price fell your wealth fell too. The payments will come as promised but their present value has fallen because they should be discounted harder. The practical implication is that you will feel bad about being stuck in a 2% coupon bond while currently trading bonds either have a higher coupon or a lower price than you paid. You lose money in the form of opportunity cost. If you try to cash out early you will be able to quantify how much poorer you are but this doesn't mean that this is the only circumstance in which you have lost.
Changes in Bond Value
Bonds are subject to slightly different risks than stocks, but in other ways they are comparable. If you hold a stock and its price goes way down, would you say you have not lost any money just because you haven't sold it yet? You shouldn't. Similarly, you should not say you have not lost money when a bond's price falls, even if you do not sell it.
Because a bond mutual fund is just a collection of bonds, at any given time its expected return and risk are exactly equal to those of the underlying assets it holds.
Bonds and stocks both have the property that they trade for the present value of their future cash flows. With bonds, especially, these prices are known fairly precisely and changes in bond values have good motivation. If your bond price goes down (whether individual or as part of a mutual fund) it is because the bond is genuinely worth less than it was.
The Dynamics of Portfolio Risk
The fact that at every given moment the risk of a bond fund and the risk of its underlying portfolio are the same does not mean the two will track each other perfectly going forward, however.
It is possible that as the fund manager changes the portfolio composition over time, she may actually lose or make money relative to a static portfolio of the underlying, but this is no different in a bond fund than in a stock fund. Any time you entrust money to an active manager you are trusting that as she deviates from the benchmark, you will like the results. VBTLX is an index fund so this isn't really a consideration, though.
Also, if you bought the underlying and held them to maturity, then your potfolio would start out with a long duration and grow shorter over time (Unless you keep buying bonds the same way the mutual fund manager does). If you do anything different with the dividends than the mutual fund does, then your risk will change over time while that of the mutual fund remains more-or-less constant.
Consider your VBTLX example. The price of those bonds fell, causing both the price of your portfolio and that NAV of VBTLX to fall. In both portfolios the bonds continue to pay out as they did before and you use the proceeds to purchase new bonds. When your initial bonds mature the value of your proceeds plus the value of your newer bonds will be exactly equal to the value of the money invested in VBTLX.
The most likely way there is a difference between your portfolio and that of VBTLX is if the manager of that fund rebalances (sells aging bonds and buys newer ones so as to maintain a target maturity). You may or may not do this in your portfolio. If you do, you will match VBTLX. If you do not, your risk will change over time.