As noted in the question, exposure metering with adapted lenses can be a major issue with some cameras. I think there would be more useful replies if the question asked for people's experiences, both positive and negative, with exposure metering using adapted lenses on different cameras, rather than asking only about which cameras work well.
Each camera has its own idiosyncrasies, beyond exposure metering, it would be a good idea try out some cameras before you buy them. Mount adapters are relatively cheap, so if you're serious about using your old lenses, get a mount adapter for each camera you want to try. However, as @Agent_L has commented, "Using an adapter most of the time is masochism." So don't be surprised if you can't get the results you want when using adapted lenses.
Focusing: On DSLRs, focusing is on plain ground glass. A light might flash to indicate that the camera thinks a target is in focus, but there is no optical focusing aid, such as the old split screens. Mirrorless cameras usually magnify a portion of the image. Some also have peak indicators, such as overlaying blue or red, to show areas of high contrast. I tend to find focusing with magnification and peak indicators (in particular, FujiFilm's system) easier than on plain ground glass.
Stop-down metering: For both DSLR and mirrorless, focusing is usually done wide open. The camera is then stopped down before the shutter is released. Metering is usually done upon shutter half-press, unless the auto-exposure lock is used.
Flange focal distance: I assume that if an adapter for a camera exists, issues such as flange focal distance and infinity focus are non issues because the manufacturer of the adapter has already accounted for them.
Camera features: Because I am addressing primarily exposure-metering capabilities, I will not describe features that set different cameras apart, such as Pentax pixel shift technology or FujiFilm X-Trans sensors.
Regarding DSLRs, the main options are Canon, Nikon, and Pentax. I have experience only with a Canon DSLR from around 2004. I have no significant first-hand experience with Nikon or Pentax DSLRs, and no recent Canon experience.
- Canon 20D. This camera had barely functional metering, even with native lenses. With adapted lenses, it was all over the place. Lenses greater than 28mm tended to underexpose. With a 16mm Zenitar fisheye, it had drastic frame-to-frame variability. One frame could be several stops too dark, while the next could have a third of the frame blown out. Changing F-stops increased the likelihood of underexposing, without completely eliminating the risk of overexposing. Upon reflection, I now think the camera simply had a really narrow dynamic range.
There are many more options if mirrorless cameras are considered. Many people use adapted lenses on micro-four-thirds, mainly Olympus and Panasonic, bodies. If you're okay with the smaller sensor size, you will be in good company.
- Olympus PEN E-PL2. This camera had no problems with exposure metering that I recall. I was mainly disappointed by the smaller micro-four-thirds sensor size. For instance, the 16mm Zenitar fisheye felt like a rectilinear lens with bad barrel distortion, not a fisheye.
Many mirrorless cameras share the same APS-C sensor size as most DSLRs. I've been playing in the FujiFilm ecosystem for quite a few years.
FujiFilm X-A5. This camera tends to overexpose, even with native lenses. If I hadn't previously used real X-Series cameras (with X-Trans sensors), this experience would have soured me on FujiFilm. I recommend avoiding the faux X-Series cameras that use Bayer sensors. (X-A#, X-T100)
FujiFilm X-E2. Excellent camera. Exposure metering is spot on, regardless of lens, unless the exposure compensation dial is accidentally switched. Main problem is poor battery life compared with DSLRs. Secondary problem is the camera is too good. I was "stuck" with this camera for so long because FujiFilm couldn't significantly improve it. The best they could manage was an X-E2S re-release. (At least they were honest about it.)
FujiFilm X-T20. FujiFilm changed so much when they designed the X-E3 that it doesn't really count as a successor to the X-E2S. The X-T20 is pretty close though. Problem is, it kind of feels like a toy compared with the X-E2 (more plastic, less metal). But image quality doesn't disappoint. Just like the X-E2, exposure metering works great. I suspect I'll be "stuck" with this one for a while. There's probably an X-T20S in their pipeline.
Many people also use Sony cameras with adapted lenses. Both full-frame and crop-sensor models are available. I do not have significant experience with Sony cameras, so cannot comment regarding exposure metering. However, I did notice some strange performance issues while an ILCE-6000.