Are there any major Las Vegas hotels -- off strip but nearby -- that DO NOT have those maddening resort fees?

No major casino hotels, but some of the smaller downtown-area casino hotels don't have resort fees (and Casino Royale, which has a good location on the strip). Here is a list of hotels in the area (many of which are either timeshare resorts or chain-brand hotels/motels) that do not have resort fees:

The following hotels DO NOT charge a resort fee.

Ambassador Strip Inn Travelodge

Americas Best Value Inn

Arizona Charlies Boulder

Bluegreen Club 36 resort

Buffalo Bill’s

California Hotel

Casino Royale

Courtyard by Marriott

Desert Rose Resort

Elara

Emerald Suites Cameron

Emerald Suites Las Vegas

Four Queens

Fremont

Hawthorn Inn

Holiday Inn Express

La Quinta Inn

M Resort

Main Street Station

Marriott

Primm Valley Resort

Renaissance Las Vegas

Residence Inn

Royal Resort

Siena Suites Hotel

Skylofts

Whiskey Pete’s

Wyndham Grand Desert

Casino Royale wouldn't be considered a major hotel but its right on the Strip and doesn't charge a resort fee. Skylofts at MGM is another on the Strip but you'll probably want to find a hotel with a resort fee with their rack rates.

Wyndham Grand Desert is about a block behind the MGM and has no fees and the Elara by Hilton is sorta kinda on the strip right next to Planet Hollywood. Both are nice - both are timeshare resorts