The rule of thumb here is: what works in one hotel may not work in another.
A second rule of thumb here is: there are at least 10 different ways to accomplish your goal depending on the hotel (not the hotel chain).
And a last rule of thumb here is: if you are not successful after spending a few minutes trying all of the options you know (or can find on the internet), stop trying and drink a beer.

What I've found to work for me is:
1 - Always look for a Source button on the side of the TV.
2 - For TVs with external LodgeNet boxes, disconnect the box from the TV, if it continues to open up the LodgeNet screen.
3 - For TVs with with some type of metal coax shields around the connectors, you can usually loosen it up by simply twisting the coax cable to unscrew the connector. This comes in handy if you need to disconnect the TV from any sort of source, such as the case with an internal LodgetNet integration.
4 - Bring a flashlight, as it may come in handy for flush-mount TVs.
5 - If it is really important to you, you can invest in an OEM remote for Samsung and LG TVs and be covered in most hotels. The OEM remotes can help you bypass some TV settings, although you'll need to know the key commands to open the engineering menus.
6 - A call to hotel services may land you someone from technical/engineering services who can help you with the TV configuration. Of course, you are not always guaranteed to have someone come up who knows anything about TVs and are simply an "engineer" in title.
7 - If you spend more than 30 minutes of time on getting it configured, it is likely time to do something else.