Printer seems not to grab paper.

That sound is either one of two common failure modes in most cases. The mechanism sounds like it works so I'm not so sure about that.

The usual common fault is caused by a worn pickup roller/separation pad, or if the rollers are in good shape it can be anything from the pad and roller being loose, up to a mechanical failure. Take a look at the roller and separation pad, especially with how old this is; if the unit has a high page count (~80-90k, 100k+) the rollers are usually worn out. It's best to change both parts as one wearing usually means the other is next in most cases (which is why most parts kits give you both parts). You can usually leave the MP tray alone unless you can visibly see they are worn or know it's seen extensive use. The reason you can get away with it is most people do not wear them down as much as the Tray 2 roller/separation pad.

The reason I suspect it's running on worn rollers is a lot of these pre-DS models are beginning to get into the PC range where it is now "high"... Because people who know better would rather run a Pro 400 M451 with this engine (or your CP variant, the difference is the Pro 400 M451 takes high yield toner but both of them run the same or at least very, very similar Canon engines which makes it a better model used) with 100k+ pages (sometimes 150-200k+) until it's dead and get another brand like Canon or Brother don't have any of this junk (HP DS/HP+/HP Smart mandatory setup).

What sold me on the Lexmark monochrome printer is the drum cost as well as a very welcome command: X and OK to run 3rd party toner :-). Lexmark is $74 for ~40k pages on the drum, and Brother is $129+ for one that goes beyond 12k pages (45k or more), but Canon still does the drum in the toner design with monochrome outside of some super cheap models. I had to transition early due to some chronic network issues plaguing the M401 network stack with modern computers and Android phones (my iPhone has never had an issue) that need regular attention. That said, I'm not putting the fight down with my pre-DS HP.
Because of how much of a disease HP DS, HP+ and the mandatory HP Smart setup series is, anyone who knows better has blacklisted HP. I either run low PC pre-DS HPs (which can be hard to find and you WILL PAY for it because they command a premium due to the lack of BS), Lexmark or Canon these days.

On some of the old designs the pickup roller issue you gave was often mechanical due to someone trying to DIY it and breaking it but that's only a thing on 90s models that had the flimsy 2X00 design. The rest were fine at least. With these, it's usually worn rollers/pads, or they came loose enough, or something is on them like a label or dirt; it still happens, but it's far less commonplace. It's a full teardown repair when it occurs.