What you are asking for is actually a standard 2-way splitter. The part pointed to in your link looks like a very basic tee that doesn't have any impedance matching. Improper impedance matching can result in some signal being reflected back from the connector (less signal continues through to the destination).
You will probably get better results with a standard TV splitter like this one: link
Note that these splitters also work in reverse. That is, the "in" signals will go to the "out" ports, and any signal entering from the "out" side also goes to the "in" port. These splitters can be used as signal combiners when flipped around.
In order for this to work, you will need to identify and label the endpoints for each of your cables. Using knowledge of where the signal is coming from, you can be sure to use the splitter in the correct direction.
The main problem with your setup is that every time you tee the signal, you are reducing the amount of signal that is available to the next link in the chain. Splitters are also known as power dividers because if you put 1 watt of signal at the input, a 2-way splitter will send half a watt (minus a little bit due to inefficiency) to each of the outputs. The next splitter in the chain would start with only say 0.3 watts (since you lose a little bit more from cable losses) and you get 0.15 watts at each output port. It gets progressively worse as your daisy chain gets longer.
An alternative way of saying the same thing (but easier) is to think in terms of dB. Each 2-way splitter is going to reduce signal strength by about 3.5 dB. Coax cable will also reduce signal strength depending on how long it is. For RG6 coax, I would estimate about 6 dB of signal loss for every 100 feet of cable.
For simplicity, let's suppose that each link in your chain has 2 dB of cable loss (roughly 33 feet of cable) plus 3.5 dB of splitter loss, for a total of 5.5 dB. At the end of the first leg, your signal would be 5.5 dB lower than where you started. After the second leg, the signal will be 11.0 dB lower than where you started. After the third leg, you'll be down 16.5 dB, and so on.
Just follow the path from the signal source to your destination TV, add up the dB of loss associated with each splitter or section of cable that you encounter, and you can calculate the amount of signal that has been lost by the time you reach that TV.
If you know the noise margin (NM) of your desired signals at the starting point of the chain, you can then figure out how much signal will be left after each link in the chain. This will give you an idea of whether or not the last TV at the end of the chain will have enough signal left over to lock onto.