Inspiron 3670, i7-8700, 3440x1440 support over Intel UHD 630?

"UHD 630" is not a display interface its a GPU.

4K requires HDMI 2.0 or Display Port 1.4

and its 30 hz  not 60 hz without a real GPU.

2560x1440 is max for HDMI 1.3

The maximum pixel clock rate for HDMI 1.0 is 165 MHz, which is sufficient to allow 1080p and WUXGA (1920×1200) at 60 Hz. HDMI 1.3 increases that to 340 MHz, which allows for higher Max resolution (WQXGA, 2560×1600)

3,840x2,160 pixel resolution at 60fps, is only possible if you have at least DisplayPort 1.2 and Multi-Stream Transport capable monitor.

You need at a minimum the GT 1030 GPU.

This card has HDMI 2.0b and Single Link-DVI so you can have

4K resolution @60hz with HDCP 2.2

It WILL NOT give you 1440p @144hz 

4096 x 2160 4K is NOT Ultra HD 3840 x 2160

https://www.dell.com/en-us/work/shop/pny-geforce-gt-1030-graphics-card-2-gb-gddr5-low-profile/apd/a9763687/graphic-video-cards

Depending on PSU limitations I have seen the gt 1030 or gtx 1050ti recommended alot for the 3670 on this forum, but can you clarify what the ultimate usage limits are for the integrated intel uhd 630?  My understanding is that for non-gaming purposes the integrated graphics are completely fine for 4k display support and video playback (HTPC and netflix/youtube type content in 4k). Is that correct? Anyone installing a gtx 1050ti (or any graphics card) in the system described by the OP would only need to do so for gaming or high end productivity?

I managed to have 3440x1440 at 49Hz over HDMI for the widescreen, but had to install the driver of the 630 chip directly from Intel, as Dell's ones are still the 2018 version. 

Yes. To use those higher resolutions you need to lower the framerate.

The easiest solution is to get a USB to Display port adapter. Then there shouldn't be any issues running that resolution at 60P.