united-states
Assuming that this is under US copyright law, this is primarily a question of fair-use. Fair use is coverd by 17 USC 107. The four factors which must be consider in every fair use decision, according to the statute, are:
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
Different factors are most important in different cases, but it seems fair to say that factors 1 and 4 are most often predominant. The court is also free to consider additional, unstated, relevant factors. In recent years, whether a use is "transformative" is often considered very important, and is usually considered under factor 1. Transformative uses are highly favored.
So let us consider the suggested use under the four statutory factors.
(1) the purpose and character of the use.
The use is commercial, and that leans against fair use to some extent. But the proposed use is quite transformative. The normal purpose of football clips is to enjoy the game, for entertainment. Here the use is to illustrated life lessons, a quite different use, and the use will apparently be accompanied by commentary. Overall factor 1 seesm to faor fair use significantly.
(2) the nature of the copyrighted work.
The work is a clip (or several clips) from a video of a sports event. This is not a creative work sartorial to a novel or a play, but it is not largely fact-based like a textbook, This factor seems neutral toward fair use.
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole.
A clip may be a small or a large part of the entire game. Or a court might evaluate the clip as a separate work. I cannot asses this factor, but it is often not the most important factor in a fair use analysius.
(4) the effect of the use upon the potential market for or value of the copyrighted work.
This is often a very important factor, and specific data is needed to asses it. If clips are presented along with "life lessons" so that a person looking just for football scenes would be unlikely to find the coach's site an acceptable substitute, then there might well be little market impact. If so, this factor would favor fair use. But this depends on specific details and what case the copyright owner might make for a market impact.
Overall, I think it plausible that a use such as is described in the question might be held to be a fair use under US copyright law, but I cannot be at all sure. Anyone planning such a use would be wise to consult a lawyer experiened in copyright and fair use issues.
In any planned use is not in the US, the copyright laws of the relent counties should be checked. Fair ue is vry spoecific to the US (and Isreal, I understand).
Would the legal effect be different if the couch simply provides a link rather than the video clip on the coach's website.
Yes it would. Linking to an authorized site such as an official NFL site, or the site of a TV station that broadcast the game in question, is quite unlikely to be copyright infringement, as long as any password/login screens are not bypassed.
