Hi,
First and foremost, I posted it to express my delight that there is an adapter that also mechanically fits my pretty good K-type sensors. 
But yes, the temperature display itself is very "nervous" - I had no time to look for it yesterday:
Is there no "Average" mode ?
Not as far as I could discover (and believe me, I did search for such an option). As for that adapter, I have to agree with your assessment - I have one myself (among a collection other similar adapters). I'd been hoping to track one down that continued the chromel / alumel wire alloys into the pins to extend the "cold junction" all the way into the banana jacks.
Needless to say, none of these adapters used chromel / alumel for the pins and, in retrospect, I guess there simply isn't any demand in this case since, unlike our Siglent bench meters with their 13 watt heat source to set up a 3 to 4 degree gradient with respect to ambient temperature, the hand held battery powered meters they're intended to be used with don't suffer such a thermal gradient issue due there being at most, only a few milli-Watts of heat energy to generate little more than a few mK temperature gradient.
The cheap TM-902C K type thermocouple thermometers I bought only draw
0.6mA 0.35mA from a 3v battery. That's little more than 1mW to generate any thermal gradient between the chip and the socket that's worth correcting for. Obviously,"cold junction" compensation for ambient is applied, probably using an on chip sensor diode to keep instrument manufacturing costs to a minimum.
I expect that your handheld digital thermometer is likely to have a similarly low energy consumption (on a par with a modern hand held multimeter) so there's little point in making such adapters using chromel and alumel pins. If I'd realised at the time that my quest for such a high spec adapter would be a futile one, I'd be about ten quid better off by buying just a single cheap Chinese adapter plug.

I did manage to acquire a connector with alumel and chromel pins, a yellow K type plug used to terminate the K type thermocouple sensor wires. The pins were actually stamped with "AL" on the negative magnetic pin and "CH" on the positive non-magnetic pin. Both pins on your adapter should prove to be magnetic assuming mine is completely identical to yours (appearance-wise they are as far as I can see).
I seem to have purchased only the one which remains unused. I guess I must have realised that it would only ever be of any benefit if I acquired a lab grade bench thermometer where a similar quality panel mounted socket was connected to a proper isothermal cold junction deep in the guts of the instrument via alumel/chromel connecting wires. The cheap TM-902C and similar handhelds wouldn't see any benefit from such a K type plug upgrade to any of my existing collection of thermocouples hence it remaining an unused spare.
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[EDIT 2023-09-22]
It turns out that I'd actually bought a set of
four of these K type plugs. I was looking through my collection of spare thermocouples today and found one with an upgraded plug already attached, swiftly followed by
then discovering the other two plugs in a resealable polyethylene bag.
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I landed up purchasing three of these cheap TM-902C meters. The first was a DoA for which I got a full refund and a spare thermocouple. Of the other two, one had the momentary on/off push button which defaulted to a ten minute power down time out, the other a slide switch which didn't have such an annoying time out feature. This was the one I chose to use, setting aside the push button one as a spare.
A few weeks (or months) later, I experimented with the push button unit to see if there was any (undocumented) way to disable the rather annoying time out feature and, as luck would have it, it did indeed have such an option. Basically, after it has booted up and is reporting temperature, it's simply a matter of holding the button down until the clock icon disappears from the top left corner of the display (about ten seconds). Thereafter until you toggle it off or leave it on for the next three months to flatten the battery, it will continue showing temperature readings. However, it remains relegated as "my spare" since it under-reads the other by almost 2 deg C. The slide switch one tracks my other thermometers to within half a degree so remains my TC thermometer of choice.