The raw speed is fine - a single user can (except for really heavy usage) do fine with 1.5 Meg. DSL, so 10 Meg. is plenty fast for typical internet use. The problem is making it all work correctly.
First of all, 10 Meg. ethernet is a bit of a problem these days. Pretty much everything will work fine at 100 Meg. and most new stuff at 1 Gig. with automatic fallback to 100 Meg. But already a few years ago I started to find devices that simply will not work reliably, and sometimes not at all, at 10 Meg. I know that sounds a bit crazy as the actual speed realistically needed by any individual device for ordinary internet use (as opposed to traditional LAN use) is 10 Meg. or less. So problem one is that 10 Meg. may not be good enough. I have also found that a marginal network connection may work OK at 10 Meg., but with 100 Meg. port on device and 100 Meg. port on switch, it may be hard to force the network to back down to 10 Meg. and left at 100 Meg. it may be unreliable.
The second issue is the cabling. Normally you need CAT 3 cabling, 4 wires (out of 8) for 10 Meg., CAT 5 for 100 Meg., and typically Cat 5e (and using 8 wires instead of 4) for 1 Gig. So 4 wires is actually fine for 10 Meg. or 100 Meg. But I have never seen 4 wire cable with twists needed for even CAT 3 operation. CL2 refers to the outer covering of the cable, it says nothing about the quality/type of the actual wires. See if you can find any additional labels on the outside of the cables that might indicate the type of cable.
There is more likely problem. Telephone wires can be done in a "star" configuration, which is similar to typical network wiring. But that is most often the case with offices where each phone jack requires a separate home run to the phone system. In most houses with ordinary phone service, several phone jacks will be daisy-chained together. That is fine for ordinary phone service, but not for twisted pair ethernet networks.
Are there ways to make use of substandard cable? Absolutely. But typically those are for situations such as "extra telephone cables in place going about 70 ft. above a 16 ft. high ceiling and another 30 ft. above a 12 ft. ceiling" and the connection to be run is for typical internet access, not a LAN, so super high speed not needed. In that case (which my brother took care of), getting a line driver (a little black box on each end of the circuit) to make it work was worth a few hundred $ and it solved the problem. Putting line drivers or similar devices in 14 rooms that are close to each other is likely more expensive than having a "cable day" and just getting the wiring done the old fashioned way.
There are also things you can do to save time and/or money in an install of this type (as opposed to a first-class office installation):
- Run double cables to single locations and put the drops on opposite sides of a wall to get two rooms with one run (i.e., 2x cable but only 1x labor).
- Use additional small switches - e.g., run a cable to a spot in 1 of 4 clustered rooms, install a 5-port switch (some are less than $10) and then run short cables to the other 3 rooms.
But I admire your choice of wired ethernet. Even if you end up providing WiFi as well (which actually does make sense so people can browse on their phones without expensive data plans), there are some significant advantages to having wired network connections.