Do you want to honor the history/sentimentality of the watch, keeping it as original as possible? Or, do you want to be more aggressive with a service and have a watch that is "newer" (manufacturer's guarantee, water resistance, etc.)? Both have their pros and cons.
If you go through the Rolex service center, be prepared for an expensive and extensive repair. Rolex overhauls start at $600 here in the US, but steps up to $800 or even $1000 on vintage calibers. This is assuming that your watch has a caliber that Rolex still has parts available for. I think there's a chance that your movement is no longer supported by them (what caliber is it?). Keep in mind that Rolex is VERY conservative when it comes to replacing parts. If they see potential functional issues with parts like the crystal, dial, or hands (moisture spotting, lume damage, scratches, discoloration), they will most likely require those parts be replaced. If the condition is good (and your watch appears to fall in that category), they can reuse them. Also note that Rolex includes refinishing of the case as part of their service, so if you want to maintain the original look - request that the case not be touched. Despite all these cautionary notes, Rolex is very efficient. They are faster than many other manufacturers on service turn-around times, pricing is fair, they do consistently good work, and I can easily recommend them for overhauls on their modern watches.
Going with an independent gives you flexibility. Independent
watchmakers can go over each of the steps needed to service your watch and are usually very willing to discuss options and advantages/disadvantages. If again the caliber still has parts readily available from Rolex, any watchmaker with a Rolex parts account should be able to do a good overhaul at exactly the level of service you want. If parts are discontinued, then hopefully the watchmaker has a stockpile of parts on hand or can get access to them. Just like the notion of buying the seller when buying used watches, you'll want to gauge a watchmaker on the quality of their work, their support for any subsequent issues, and not just price. Vintage repairs through independents can take longer (think several months vs. several weeks for Rolex), but you get a much more personal experience.