It may seem frustrating, but we will be able to make it send every function; but you do need to plan on spending a half hour or so of work. So far, by showing that setup codes 11178 and 11758 work, you've established that your TV responds to the NEC1 IR protocol, device 4, and that the volume and channel groups are working. The Comcast remote has 8 setup codes which send this IR protocol. However, all of them send the same function number for Power (00053), so I'd like you to recheck if 11178 or 11758 cause the power to shut off.
I can't tell from your description which buttons do or don't work. Statements like "It will actually pull up the input options, but it won't allow you to select a new input." aren't useful to me for two reasons: 1) I don't know what "pull up" or "select" means, and 2) or what button you pressed to get the action.
There are two courses to take here. One is to try all 256 possible EFC numbers (00000 through 00255), while watching for some reaction from the TV. This is quicker than it sounds, but for example, the OK/Select function will only be active if a menu is already displayed. So some EFCs may have to be tried twice. See post #7 for instructions on how to try EFCs. Don't forget to write down the EFC number when you get a reaction!
The other way is quicker if it works, but isn't sure to provide an answer. The possible setup codes that may already be correctly programmed to run your TV are 10030, 10178, 11178, 11240, 11265, 11331, 11758, and 11993. Don't bother with 11993, it is a duplicate of 11265, but with some additional functions for a combo TV/DVD. 10178 is a superset of 11178, so no need to try both. I believe that all of these will control Power, Volume +/-, and Channel +/-. I expect 10030 to send a different signal than all others for Mute and Input (TV/VCR). The cursor buttons are different among 10030, 10178, 11265, 11331, and 11758. OK/Select has three values (00036, 00164, and 00251); 10030, 10178, and 11240 will provide these. Try each of these setup codes, noting which ones work in a given situation, as well as the button which caused a reaction. You may need to use the OEM remote to, for example, bring up the menu to see which setup code can issue a OK/Select signal. If yo u need to use multiple setup codes to cover all the needed functions, that's OK--we can use keymoves to get one setup code to do everything.