Is there any hope for my printer, or should I just buy another kind?
Yes, but probably yes for another printer too - though not necessarily another kind. Note the OJ 4215 was an entry-level, budget priced multifunction printing device from the start - which was in 2004. You don't say how many years ago or how heavy a printer user you are, but I note it received a good review from PCMag
the 4215 offers better output quality than most AIOs in its price class, which makes it a reasonable choice.
All printing devices are very mechanical, and therefore highly subject to mechanical failures and general "wear and tear". You say you have generally been happy with this machine, I personally like HPs and don't think you have a better chance of getting a more reliable device simply by changing brands. If you have had that machine for several years, and your biggest complaint is it does not grab envelopes, I'd say that HP product has served you well - considering your initial investment.
I say yes for there being hope because grabbing the paper is a common and I suppose inevitable problem for all printers, copiers, paper feeders, typewriters!, and the like - pretty much anything that uses rubber rollers and wheels to grab and move paper. Over a relatively short period of time (depending on how often used and paper quality - dust and roughness) these rubber parts become worn, smooth and sometimes glazed and slick. The rubber needs to be scuffed up so it can get a grip. If you are skillful, have steady hands, and can figure out how to access all these paper moving parts, you can use a rubber rejuvenator
which will clean and "dress" or "condition" the rubber so it can get a grip. But you can only do this one or two times with success because you can't stop the rubber from wearing away from normal use - and simply getting hard from age - that's just the "nature" of rubber.
You could take the printer to a shop for a thorough cleaning and rubber conditioning, but the labor costs may not be the best economical choice compared to the costs of a new device (and the lower prices of ink on the newest printers).
As far as envelopes specifically, what I eventually had to do on both my last printers, and expect I will have to one day on this new one is to put gentle pressure on the envelope, very gently pushing it into the grabbing mechanism just as I click to print. This pressure was enough so the enveloped was grabbed securely and immediately at the start, where the problems always tended to be. Once grabbed correctly, it flowed through the printer just fine. That was fine for me who printed one or two envelopes a week. If you do mail merges or otherwise print many envelopes a day, holding your hand up there all day would be impractical. That said, if you regularly print stacks of envelopes you should consider a printer more designed for commercial, or full time office use.