The public did not know about X-Com, but the goverments did. The goverments are spending their money on troops and equipment and they DEFINALTLY wouldn't let a SWAT unit get in the way.
In X-COM, did the majority of the people in the world already knew about the aliens and X-COM? If they did, I might understand about the police being ordered to not interfere, but otherwise, the police probably won't know very much, and probably send in a small force to deal with the problem, hoping the goverment doesn't find out. If they found two seperate forces fighting in their city, alien or human, they may take what they feel are appropriate measures.
As for why X-COM is needed: X-COM is an entity with the troops and equipment that are required in order to eleminate the alien menace. The police and SWAT, while quite good against most armed criminals, are not usually employed against military forces, and usually lack what is required to counter such forces sufficiently.
Now, as for making it easier, it probably may do so, but then again, the police and swat don't have to be particurly very good to slow down the aliens or X-COM a little. They are basically slightly stronger civilians with weapons and no armor.
The saving issue in this title probably can't be elminated altogether, but two or four less reloads can be quite good, depending on who you are.
1 policeman in the wrong place at the wrong time would be fine, but I would seriously think that a whole SWAT team would be pointless if they couldn't do anything 2 them, and if they were stronger then it'd make the game easier. If there is a major incident, for instance the Iranian Embassy is a famous one, the police were told 2 secure the area no nothing else. SF's did the actual work they were instructed 2 stay out the way without actually knowing that it was the SAS since they aren't supposed to officially exist.
IMO a couple of peeps with guns is fine, but a SWAT team would be lame.
If peeps have 2 keep reloading it is a testimont 2 how hard the game is. It may be the only way some peeps can progress in the game, so we shouldn't restrict them but reward them with braggin right 4 not saving, a la using Ironman.