Reading through this thread there is plenty of good advice about your original query for a 'retro' bike that will act as a fast, comfortable tourer. Some believe you may want to keep up with the 'chaingangs' of local clubs, most of whom ride carbon bikes (usually black to match their outfits
).
The issue there is that cyclists who enjoy 'touring' style riding will not enjoy the group riding style that local club runs, that in the main will be club members who like to race occasionally, can entail. In saying that you may be lucky that your local club has a 'casual rider' run that you can join.
In my experience as a lifelong 'tourist', now 68 years old and having more time on my bike(s), solo riding, or occasionally in a very small group (3-4), is the essence of tour cycling. You go where you please, when you please, at your own pace. For that you, quite rightly, want a comfortable bike.
The above is a scene setter to impart my advice/experience from trying to achieve the result you are seeking. Admittedly this has led me to buy multiply bikes, but any one would be suitable on its own.
You can buy a relatively modern touring bike, I have a 2018 Holdsworth Stelvio, Ultegra 11 speed, disc brakes, Brooks saddle (the best), mudguards, that I use all year round. This is a welded steel bike (from Planet X) that I bought on ebay, nearly new, for a good deal less than new price. It is not the lightest, but rides very well, ride quality up with my other bikes. You can pick these up for your price target on most used sites.
I have a 1983 Holdsworth Mistral, the epitome of a 'retro' tourists bike, 531 frame, mainly period components, Brooks saddle (did I mention comfortable), but importantly, as others have said, with modern tyres (25c Schwable Laguno's). This is my 'down memory lane' bike, downtube shifters (Suntour Powershift, the best) and all. Again purchased on ebay some time ago. You can, again, pick up similar (CB Majestic, Dawes Galaxy, Raleigh Classic/Royal, etc.) for around your budget, if you want pure retro.
Another route mentioned, buying frame/components/wheels separately, I have done for my third example. I had bought a set of wheels off ebay, as spares, newly built 700c Mavic Open Pro rims, db spokes on to 20 year old Hope TiGlide hubs, with new bearing fitted. I subsequently found a Shimano Ultegra 6603 groupset (10sp rear, 3 chain rings !), also on ebay, that no one else had bid on. When it arrived, true to the description, it had had little use and was in excellent condition. I just had the crankset polished to remove obvious scuffing. Now that I had all these parts I decided to look for a frame to join them all together. As mentioned throughout this thread, I took my time to find exactly what I wanted, in a condition I was willing to accept. I ended up with a (rare) Raleigh 531C from 1988, 130mm rear spacing. This is probably close to your requirements for about your budget (not that I am selling it, just giving an example !). A 531C frame with modern(ish) groupset, i.e. flight deck levels for brakes/gears, 30 gears (52/39/30 ft, 12-28 rr), sealed bearing wheelset with retro modern rims running on Continental Grand Prix 4000 23c tyres, Brooks saddle (there is a theme here).
So, to conclude, after my long winded diatribe, you can achieve your aim, just take your time. The bike of your dreams is out there, you just have to wait until your paths cross !.
I have attached pictures of these bikes to show off !