Some of the most recent Sony camcorders and digital cameras are beginning to have remote control ports added back to them, although I don't think that they are LANC (they might actually use the LANC protocol, but not standard LANC jacks, but I don't know for sure). For example, the Sony HX50V digital camera has a MULTI port on it, and the RM-VPR1 wired remote control connects to that port. The remote has zoom, Record start/stop and power on/off on it, and I verified that they all work with the HX50V.
To find out more, go to the Sony.com site, then search for
RM-VPR1 (a wired remote)
VCT-VPR1 (small tripod with wired remote)
RM-VD1 (a wired remote that works with some older discontinued Sony Handycams)
For each of these devices, after you find them on the Sony.com site, click on the Compatibility tab to see which cameras and camcorders that they work with. For example, the RM-VPR1 is said to be compatible with: "DSC-HX300, DSC-HX50V, DSC-RX100M2, HDR-CX220, HDR-CX230, HDR-CX290, HDR-CX380, HDR-CX430V, HDR-PJ230, HDR-PJ380, HDR-PJ430V, HDR-PJ650V, HDR-PJ790V, NEX-3NL, SLT-A58. The following Alpha cameras support Shutter Release only: SLT-A65V, SLT-A77V, SLT-A99V."
Most, if not all of the camcorders and cameras will have an HDMI port that you can use for monitoring purposes. For example, the HX50V has a micro-HDMI port on the bottom, near the tripod screw (not exactly convenient for your purpose.) However the HX50V does not have a thread on the lens to accept adapters and wide angle lenses. It appears from the Sony site that the HDR-CX430V camcorder has filter threads which can accept a wide angle lens, if the 26.8 mm (35 mm equivalent) that it can do without an adapter in movie mode is not wide angle enough. It also has a micro-HDMI port. There are probably other Sony camcorders that would work, and which would cost less than the HF G20.
Another comment is that the MULTI port (which appears to be micro-USB) on the HX50V is also used when connecting a USB cable to charge the battery or to download from the memory card.
I have a Sony HDR-CX550 Handycam that has filter threads and a remote port (it uses the RM-AV2 wired remote, which I have and is similar in appearance to the RM-VD1 remote.) It is a nice camcorder and also has a viewfinder, along with the LCD, but I don't use it much anymore, as I now use a waterproof Sony GW77 and a host of Sony digital cameras (HX5V, HX9V, HX30V and HX50V) to shoot video. I'll probably put the CX550 on eBay some day, but it might not work for your needs because the port that the remote connects to is also used for analog A/V output, and you can't use the remote and analog A/V output at the same time. However, the CX550 also has an HDMI port, which would work for your needs if you were planning on monitoring via HDMI.
One issue with all of my suggestions is that you mentioned the ability to control the camera from 25' or more away, and all of these remotes have cords shorter than that. However the RM-VPR1 remote has a removable cord that has a weird connector on the end that connects to the remote, and a micro-USB on the other end. I imagine that there are micro-USB female to male extension cables available from third parties that might work. There are certainly USB to USB extension cables, and you could add USB to micro-USB adapters, but at some point that would become somewhat "kludgy".
I hope these suggestions give you some options to work with in your price range.