I'm not in the US so a lot of this probably won't apply but here it's very difficult to get a quality upright for anything less than £2000 (around $3100.) What you will find at that price point are Yamaha B1s, Chinese makes like Steinmayer, Waldstein, Weber, maybe Kemble. I paid roughly £3000 for my (used) Yamaha U1.
When I was searching for pianos I did see used Baldwin uprights for around £500 which could suit your purpose. However they were from private homes and if I were you, looking to spend $1000, (which is a lot of money but probably won't buy you a whole lot of piano) I'd save up until I could buy something from a dealer that was warrantied. My U1 would equate to around $4000 but it's grey market so you'd probably find them much cheaper in the US. You could also look at Pearl River models in that price range, used Kawai, etc - they'd make much better pianos and you'd be much happier than if you were to buy a so-so piano for $1000.
I've bought a piano from a home (steinway 40) and a $50 spinette from Salvation Army resale, I do the inspection myself. There are dozens of brands from the fifties and sixties that can be had for $1000 or less, but the quality brands tend to be bright. I like Baldwin Acrosonic, Baldwin Hamilton, Sohmer, Mason & Hamlin, Steinway, Wurlitzer, Cable Nelson, Grinnell Bros. You won't like any of these, they are bright pianos. some of the cheap Wurlitzers are a little dull sounding but the scales are not all that good. Cheap console pianos have more missing dampers on the treble and the strings don't go above the hammers on treble much. The short string scales are harder to tune by ear in octaves, the overtones don't match up like the longer string units. I tuned the bargain Wurlitzer in the fellowship hall downtown, to an extension speaker off the Allen organ. You hear the fundamental over 61 keys and 24 pedals, I can tune to it accurately. A fifties baldwin hamilton, I can tune it effectively in octaves by ear from the pitch stable bass notes.
The 2007 Yamaha 44 studio piano at church is dull and insipid enough for you, but the bass is puke sounding also. Before Yamaha, there were Everetts, that tended to have a good action but a dull and boring tone. My mother's 1947 Everett 40 console held tune very well, but the tone was really understated.
For superior bass and a dull tone, at the $50 to $100 price, look for a 1910-1938 upright piano, the 44 to 48" tall ones. The heavier the better, the more wood they stuffed in, the duller the tone is. These uprights tend to weigh 400-500 lb and piano movers charge extra to move them around. Brand name doesn't matter much for quality, they were built from kits from wholesalers before the depression. Some had great actions some had ****, you have to sort out the 100's of brands yourself by playing repeated notes to see how fast they are, and play soft to see if the action is consistent. Most of those old uprights hold tune like the pins were welded in. You'll see Kid rock's band (at Graceland, PBS) playing a plain twenties upright for the bright tone, whereas the pricier uprights at Salvation Army resale have 100 lb of decorative woodwork on the front and a much duller tone.
Watch out for laminated pin blocks if you like dull tone, I find my 1982 sohmer 39 has a laminated pinblock and is very bright sounding, whereas the 1941 Steinway 40 is a little less loud but the tuning is very stable. The 41 Steinway has a solid maple pin block, after I got the tuning up to pitch (6 passes) in 2011 the pitch was okay until 2014 winter. the 82 Sohmer needs tuning yearly or more if I don't aircondition in the summer.
If you want to buy a modern import for dull sound, be sure to check the consistency of the action playing very soft notes. I played an imported nineties "wurlitzer" (general music) at a students home, the soft touch was so inconsistent as to be unusable. The 195? $50 Wurlitzer spinnet from DeKalb IL I bought last month at SA, the action is very consistent. The $50 spinet is dull enough sounding for you but the bass, as all spinets, is nothing to write home about.
I'm glad you other posters live in London or the eastern metroplex where you have hot and cold running techs. The tech that worked at the Steinway dealer here in the eighties, he was tuning my Sohmer 1/4 tone flat, and his prescription for the pin that went flat in two weeks after he left was a damp chaser unit. Totally useless that guy was, the damp chaser did nothing but evaporate quarts of water here 1/2 mile from the Ohio river in a non-air conditioned home. I got some double sticky tape for the loose pin hole last year from Steve's Piano Service. $5. (I got a $800 "scratch and dent" discount on the Sohmer, and I lived 250 miles from the dealer so he told me there was no warrenty at the price he gave me). And the tech now in this town that advertises on classical FM radio , he tuned a 1920's upright for a minister I know, and quoted them $2000 to "completely restore" it. I played it, it has been lightly used and I wonder what issues he was worried about. Nothing I could detect in 10 minutes playing. My 1941 Steinway has more issues than that 1920's upright. For a $5000 discount from the listed 41 Steinway on e-bay (good reputation) in NY state, I'll deal with my action issues as I learn how.
I'll check my old pianos myself before purchase, thank you. Speed of repetition with two fingers on one note fast enough for me, hammers and dampers not worn in the middle, no mouse chewed damage, no pins a lot flatter than the others, no sticky keys, no missing strings, or missing treble strings only for a big discount, no hammers or action parts warped by water exposure, no action parts wiggling from side to side. (The $50 Wurlitzer spinet I bought last month is not that fast, but fast enough to put in a trailer at my summer camp that is un-heated or cooled summer or winter and might develop roof problems any time.) On a used piano candidate I do allow not working pedals, usually something unhooked in the last move, missing key tops, and good sounding spliced strings. All these minor problems I can fix myself. I allow vile, but uniformly flat (treble) tuning, that is to be expected for the best bargains.
Or you can throw money at a dealer or flipper. The dealer tuning adds $500-700, and that pretty story the good looking salesman tells is definitely worth $2000 more. That is how my church ended up with that ****y 2007 Yamaha studio from "the dealer with the beautiful green eyes" as one happy Yamaha purchaser (a woman) put it. The Yamaha studio sounds like it is under 6' of mud, I won't even sing with it.