Car tyres tubeless or runflats ?

The existing stressed tyre looks like a Primacy 3 95W.
If correct, a like-for-like replacement target price stating abbreviation "RFT" would be around £145 INCLUDING fitting, random example as base comparison:
ractyres.co.uk/mic…y-3

or if you don't like abbreviations and specifically want to see "run flat" in description then £160 each PLUS fitting, random example as base comparison:
tyres-guru.co.uk/pro…528

All car tyres are tubeless.

Both tyres are tubeless.
Runflats have a stronger sidewall on the tyre so you can drive at reduced speed to a point of safety. However they can give a harder ride and if your just pottering around unless you have tyre pressure sensors fitted you might not know you have a puncture. Also you are not supposed to repair runflats.
It's a lot easier to tell if you have a puncture in a runflat as you will visually see a deflating/deflated tyre, and as long as the puncture meets certain criteria you can have the tyre fixed for about £20.
IMO the best combination is not to have runflats and to have a spare wheel, even if it is a space saver.

Looks like they have been run for a while under inflated too as well as kerbed

The post citing QuikFit + Halfords + Blackcircles doesn't (obviously) indicate if the prices are for run-flat option or not,
so probably best to assume the prices do not apply to run-flat.
Same post indicates the previous assumption is correct (Michelin Primacy 3 205/55 R17 95W) and where the sidewall of the existing tyre should state the identifier (manufacturer code) that will literally identify if run-flat or not.

Costco is acknowledged to be a competitive tyre supplier and Costco lists the run-flat version and non-run-flat separately and openly states the identifier on each applicable webpage:

tubeless Michelin 205/55 R17 95 (W) Primacy 3 run flat (ZP) identifier 390965:
costco.co.uk/Tyr…965


Compare to same but without run-flat:
tubeless Michelin 205/55 R17 95 (W) Primacy 3 identifier 564578:
costco.co.uk/Tyr…578

RFT tyres are a very marmite subject. I used to commute long distance and I had RFT tyres on the car (BMW 1) for about 7 years. They are expensive, noisy and the stiff tyre wall make the ride harder. I often thought about swopping them out for conventional tyres which are cheaper and quieter. That was until one winter afternoon in appalling weather in the middle of rural Wales I picked up a nail. TPS sensed the pressure loss and I had 60 miles to go. At the required reduced speed I got close to home and found a small indy tyre garage who repaired the tyre. (Worth knowing that many big tyre chains will not repair an RFT due to the possible heat damage to the tyre walls.) Anyway, that one experience made the expense and noise very worthwhile.

Cheapest I have found is at Kwik-fit: £150 with Sunday fitting date
kwik-fit.com/tyr…373

Other costlier:
Halfords: £157 Tuesday fitting date
halfords.com/tyr…202

Blackcircle:£150 with Tuesday fitting date
blackcircles.com/cat…652

Is Kwikfit ok/reliable place ?

Kinda weird that OP should present list of apparently 3 cheapest tyre suppliers that are all about £5 more than the earlier suggestion, and where those 3 suggestions do not state run flat.

So to repeat earlier and cheaper suggestion that specifically states "run flat"
ractyres.co.uk/mic…y-3

or same price / supplier if you want to roll with ASDA
asdatyres.co.uk/mic…y-3

and eTryes (all part of Colewood Automotive ecommerce)
etyres.co.uk/mic…y-3

but all marginally negated by the 1% cashback if choosing to pay more at ATS