I checked several 70 cameras I have. I could not get them to malfunction using a cover with the raised tabs like the ones on the inside side of your cover.
I used a full 100 ft load of film. Both daylight loading reels were the metal ones like Kodak uses, R-90. In all the years using these cameras, I've never encountered any speed problem relating to the cover. If you are using any reels other than these, plastic ones, or severely bent metal ones, that may be the problem.
A metal camera spool with a 3.615-inch (92 mm) flange diameter and a 1 1/4-inch (32 mm) core diameter. Square hole with single keyway in both flanges. Center hole configuration aligns on both flanges. For 100 ft (30 m) film loads.
There may be some old reels floating around, maybe that came with your camera, that have a square hole on one side and a round hole on the other. They could be marked Bell & Howell, Kodak R-90, Kodak, Cine Kodak, or DuPont. The newer reels have square holes on each side.
However, what I think is wrong with your camera is, simply, it needs to be taken apart and cleaned, removing the old dried oil and grease.
Your camera most likely sat for decades without being used or oiled. The instruction book recommends oiling it after a month of non-use. So the oil and grease in the camera dried up over time. Adding new oil got it running, and it sounds good without film. But the dried oil and dirt is still in there, gumming it up and putting a drag on the mechanism, not running at 100% as it was designed. The shaft for many of the gears sit in brass bushings on the 2 plates, and these get gummed up from the dried oil.
You load the film, wind the spring up for full power and the camera starts out running at speed. The film traveling through the sprockets and gate is putting a load on the mechanism. As the spring winds down it's loosing it's power, and the weight of the film is building up on the take-up reel, dragging the take-up clutch. As the un-winding spring gets weaker, it can't drive the mechanism at speed anymore, the camera slows down and stops when the spring is fully un-wound. If the camera is in top shape, there's a 'buck tooth' on one of the gears called the 'stop gear' that stops the camera before the spring completely un-winds. This prevents the camera from loosing speed before the spring is so weakened it can't drive it.
Too me, it sounds like the governor in your DA needs cleaning and re-oiling. Also the main drive off the spring 'motor' needs to be taken apart, cleaned, greased and re-oiled. The repair manuals suggest opening and cleaning the old oil off the mechanism if the camera is not running at speed.
I've had this happen to several Filmos I have.
Get it cleaned if you want to use it, or look for a later model camera - DL, DR, and gamble again. Use the lenses from your DA on the newer model if you want that old lens look.
Another thought, the spring could be loosing it's power and needs to be replaced. However, I've only heard of the spring breaking.
Search for old posts in the forum here. There's other discussion about this.