They are correct. A PC's speed depends upon the processor, ram, hard disk & motherboard it is equipped with. A Ethernet (wired) card or a wireless card will not slow down a PC unless you are browsing the Internet, in that case if all three PCS are browsing at once, Internet browsing speed would slow down as the usage increases among all three PCs. The speed of Non-Internet Computer Usage would not slow down. Now if you are just working in something like Microsoft Word (non-Internet based) & the PC is slowing down, then there may be a problem with your Ethernet (wired) or wireless card causing the PC to accept all data being sent out over the network, even if it is destined for the Internet, therefore slowing down your PC because it has to process that data, taking up valuable processor availability & RAM space. In the case of the prior I would take you PC somewhere to be looked at.
Also a question I forgot to ask is, Is your main PC Sharing your Internet connection or do you have a wireless router doing the sharing for you & the main PC is pulled it to it or connected via wifi?
If the answer is yes, what would be your reply. We just hooked up wifi for my nook tablet but our main computer had Internet through comcast however now the computer is so slow it barely will pull up the Internet.