"Are checks worth a whole lot of money just because they can transmit money?"
Yes. A check is an instrument provided by a bank. And banks are worth a lot of money according to the stock exchanges, last time I checked.
But the bitcoin network can operate even if one bitcoin is worth about 10$.
Yes and no. For the moment, for sure. The point is that the "price of a bitcoin" needed to do a certain number of transactions, with a certain amount of time in between them, is exactly what the monetary formula gives you:
P x Q = V x M.
P x Q is the total price (in bitcoin) of all goods and services bought with bitcoin in a certain period, say S, the total sum of bitcoin expended to buy stuff.
V is the number of times on average the same bitcoin participates in a buying action of such good or service in the given period (say, a year).
M is the amount of bitcoins in circulation.
Now, V is the inverse of the (harmonic) average hold time T of a coin. Indeed, you do not immediately upon reception of a coin because you sold something, buy something else with it. If you sell a car today against coins, you will probably not spend all that money in the evening. So all coins are held a certain time on average in between two interactions. A dollar is held a few months on average for instance.
Let us assume that a coin is held on average half a month. That would put V to 24 (if the period is a year). Now, with finally 20 million coins in circulation, that would bring us to a total sum spent on goods and services of 480 million bitcoin.
Every year, one will buy for about 480 million bitcoin goods and services under these assumptions.
If your bitcoin is worth $10.-, then that means that every year, for not more than about $ 5 billion goods and services are bought with bitcoin.
If you want to buy MORE stuff, then there's no alternative but for the price of bitcoin to go up (or for the holding times to go below half a month).
If you want to buy 20% of all of the black market (estimated $ 300 billion, so $ 60 billion bought with coins), then a coin needs to be at least 12 times more expensive, or at least $120 a coin. (under the assumption of a spending every 2 weeks).
So the price of a coin cannot be arbitrary low if it has to buy large volumes of goods and services.