There are some misconceptions here. First off, as a rule, the media doesn't "pay" for access to sporting events. Occasionally, it is necessary to purchase a ticket just like the paying public, but these occasions are few and far between.
What media credentials (we don't use the term "press pass" much anymore) allows you to do is get access to the field or baseline or holes in the glass, etc. As a newspaper photographer you will receive the creds at the paper and when you go to the game, you will need to have these creds prominently displayed (usually a big tag on a chain around your neck). You may or may not have to wear a photo vest provided by the venue.
Why there is special access for the media is that the venue appreciates what the media can provide them, which is publicity. As a courtesy, they provide access free of charge and allow the media to photograph the game from areas that give the best possible images.
With the exception of small, hometown games, media creds are not that easy to get. Bear in mind, creds give you access to the field, baseline, etc., as well as press room areas.
Most venues for professional games have their own creds so any kind of "state press credentials" won't help. As for that, where I live, there are no general creds issued by the local police or state-wide organizations at all. These credentials are requested by the media outlet and, in most cases, must be made by the photo editor or managing editor of the media outlet, sometimes months in advance. Depending on the venue, they may just say that the media outlet isn't big enough and deny credentialing totally.
On the original question, the reason venues refuse to allow "big" lenses is due to the photographer being able to post photos without their consent on the web. Right or wrong (personally, I think it's silly) that is what they are worried about. Since the security officers seldom know what constitutes "professional" gear, they just make a designation, such as no six-inch long lenses, and stand on that. Obviously, if you can talk your way around a security guard, so much the better.
Jerry