Welcome.
Did you use the service manual?
"Clean the mode switch" has become a stupid meme like "clean the heads". Only people who don't know a damned thing about VCRs are suggesting that. Cleaning a mode switch, and cleaning heads, is not a fix-all for whatever ails a deck. Those should be specific solution to specific problems.
What happens is that disassembling a deck to even get to a mode switch can complicate matters. For the untrained, putting a VCR back together properly is no small feat.
Another truth about VCR repair is that it helps immensely to have a proper working unit to compare against. Not necessarily the same exact model, but one from the same family (not the same brand, but lineage/family).
When a unit fails to pull the tape back, it has an issue with the brakes (or related). I've discussed this in other posts (so has latreche34). I only come across this problem less than once per year, so it's not that common. There can be multiple causes of this.
This is important: Have you ever seen this VCR work properly before? Or was it some sort of thrift/eBay find?
The 3500 is honestly too old to worry about. I'm already reluctant to recommended the 3600/4600 for a non-TBC JVC deck, vastly preferring the 3800/4800 units. 2nd would be the next "model" (29U, 29xx, 59xx). Some of those pre-1996 JVCs were really grouchy, and the x500 decks were the "missing link" between ancient worthless S-VHS and recommended S-VHS. Those x500 exist in "caveat land", use with caution, and you really need to know about your gear.