It's possible to sell a t-bill out of a TreasuryDirect account, if you have to. You get the market rate (so you're no longer guaranteed to get face value) and there's a fee. (Although it's cheap compared to getting a broker to do it, it can be a large percentage of the money you're getting, especially if your emergency fund is small.)

Still, you're generally right, and that's why I only keep half my emergency fund in the t-bills. For a large class of emergencies (such as losing a job), you don't really need the money right now, you just need to be really, really sure the money will be there when you do need it. You need to make a judgement call about the kinds of emergencies you might face.

There are two problems with using a mutual fund. First, even a low-cost fund can eat up a good bit of the total return, because the interest rates on cash are often low. Second, you lose some of the biggest advantage--the safety of the investment--because you no longer own a direct obligation of the US government. Instead, the mutual fund owns a direct obligation of the US government and you own a share in the mutual fund. I doubt if that makes much difference in practice, but if you don't care about that tiny increment in safety, you'd probably get about the same return in an on-line bank.