Ultimately, it comes down to use and architecture.
Architecture
Does the system handle "any sport"? Is the idea that you put on your architecture astronaut hat, and build a generic system that can handle any future type of sport that may not even exist today?
If so, obviously having dynamically-named tables is a huge pain, so it would make sense to have a schema that supports n sports, if required.
That said, I have a very strong bias against this approach: this is almost always more work and leads to poorer results. Making a separate UI, schema, etc, for each sport will ultimately lead to better user experience and easier to maintain code, even if it does mean some superficial amount of duplication (how to avoid/minimize this is a separate question).
How do you handle players that play multiple sports? Do they get two entries (eg, you treat as different people) or are you trying to do something specific with them?
Use
So let's assume you don't do sports dynamically (eg, if someone wants to add a new sport, it requires development effort to add it).
Is there ever a time where you're displaying players (or any other object you mentioned) from more than one sport at a time?
I could see this for a search function, where you could search by player or team name (regardless of sport), but beyond that I can't imagine many use cases.
If you never need to do this, then your approach is perfectly fine. You can stop reading here.
Alternative Schemas
Views
I'm a fan of KISS. In 15+ years of software development, I continue to fall back to the philosophy "build the simplest thing that works".
So my initial reaction, assuming a cross-sport search function is really the only use case, is to create views:
SELECT PlayerName, 'NFL' as [Sport], TeamName FROM NFL_Players JOIN NFL_Teams ...
UNION
SELECT PlayerName, 'NHL' as [Sport], TeamName FROM NHL_Players JOIN NHL_Teams ...
UNION ....
Of course, if you add a new sport, you have to add to the view. It may also be useful to include other common information but really that's dependent on what needs to get shown.
I'd try to keep all the sport-specific stuff in the View definition, so the search code doesn't need to have much or any specific code (besides maybe knowing how to link to /nhl/players/player-name vs /nfl/... or however your app does that).
Table Inheritence
Table inheritance can work, but is pretty complex. I don't have a ton of experience with it, and in fact, I think every time I've been involved in evaluating it we ended up doing something simpler (like I'm suggesting here).
So personally, I have yet to find why this would be useful, but maybe there is a convincing use case (that I don't know of) that justifies the complexity (eg, Table inheritance solves the use case better than any other solution).
Separate tables for sport-specific attributes
You could do a single players table which has attributes common to all players of all sports, and then another set of tables like nhl_players_details that contains a playerId and columns with additional info on the player. If there are a ton of common attributes, or you have many uses of "all players from all sports" then this may make sense.
Key value pairs for sport-specific attributes
Completely alternative approach: have a players table (again, with common attributes like name) and then a player_data table that has PlayerId, Sport, Attribute, Value. The attribute names entered would be sport-specific. This allows you to essentially add new attributes without modifying the schema (your code would still have to know to load/display them of course). The drawback is you lose some integrity: value would typically be a string field, so your app code would have to be resilient and handle potential failures converting the string value to a specific data type (like integer).
This concept can of course apply to Teams, Games, etc.
Best Answer
Why not let the text search engine like lucene or sphinx do all the work? I use Lucene and have found it to be very good at searching multiple tables and fields for whatever you set as a target.
You don't say what your web front end is written in but both of these tools can be used with most languages.
You will be adding a lot of code to maintain in the triggers that watch for created/modified/deleted in all the tables. Wouldn't it be easier to add the option for admins to mark a column as "to be searched". Then write code to add this column to indexed fields?
I see that Sphinx has "offline index construction and incremental on-the-fly index updates" which would seem to allow you to add a field, rebuild the index offline and then bring the new index up.