Some consider the combination of 4xCompression rate/1xRebound rate (over standard) to make the Bilstein ill suited to UK roads (due to uneven road surfaces); I personally think a well set up Bilstein car with factory springs rides just fine. MX5s are not boulevard cruisers, and the S-Specials are supposed to be the sportier variant.
The factory Bilstein shock has an aluminium lower spring perch about 10mm lower than the standard Showa shock, although the springs are a very similar spec to standard. The overall effect is to lower the car by about 10mm. Mazda then fitted modified track rod ends to Bilstein cars, to correct the steering geometry and reduce the bump steer that would result when lowering a car. All very carefully thought out.
All OE replacement shocks are intended to replace the Showa shock absorber, not the Bilstein. KYB go so far, in the US, to state that their popular AGX line of shocks should not be fitted to R-Package Miatas (the US equivalent to the S-Special), I suspect due to this height change; the car would be raised slightly, and the steering geometry would be…??
Bilsteins do not slowly “go off” like standard shocks; when they go, they go, so possibly there is nothing wrong with the shocks (look for a shock that looks wet, if a seal has failed). With a coilover kit (technically, all MX5s are on coilovers, but people really mean adjustable coilovers), you have the option to set the height to what you want; you could set it to your existing height.
I’ve had P5 Puredrive coilovers since 2008; in that time, I have only had cause to touch the height setting once. The car is more or less at the height I got the car at, which was on Bilsteins and some unbranded Japanese lowering spring, which means it wasn’t lowered all that much.

Shocks that come with rebound adjustment; fine if you are lowering the car, or are wanting to switch between track and road use, but its another feature that probably 80-90% of users never touch, once they have decided on the initial settings. But, beware of cheap coilover kits masquerading as uprated shocks without rebound adjustment; these are just a cheap strut cartridge for a Nissan or something, inside a Mazda specific tube. They can’t control the spring when lowered, and so some sharp witted users will complain of a “bouncy ride” or similar. The Tein Basics are uprated.
Another factor to consider when changing the shocks is how good are your springs; MX5 springs tend to crack at the top when aged, and the labour needed to change them is the same as changing the shocks. If you are labouring yourself, the extra work doesn’t really matter, but if you are paying someone else to do the work, worth bearing that in mind. Replacement springs are relatively cheap.
If your springs are in good nick (ie. NO rust), and you are otherwise happy with Bilsteins, I would consider replacing like for like, and retaining originality while you can. An alternative approach, if you can afford for the car to be off the road for a period, is to get Eibach UK (Bilstein UK) in Leicester (KBSS Ltd, Unit 3, Swannington Road, Broughton Abbey, Leicestershire, LE9 6TU. Tel:01455283407) to refurbish the shocks back to factory spec. They used to charge £55-60 plus VAT per shock, £288 for all 4 shocks, compared to £455 for 4 new Bilsteins from MX5parts. Apparently it take a fortnight , depending on how busy they are.
I have found, in the past, that provided you renew split bump stop gaiters, these shocks can be very long lived. I am not sure about the longevity of any of the other aftermarket shocks, with the exception of Koni and KYB, because they are essentially boutique shock manufacturers by comparison, and can’t possibly put anywhere as near the amount of development as Bilstein did.
Your choice will probably be determined by how you can do, how long/how you use your car and the state of your springs. New Mazda springs will probably be around £200, and bumpstop/gaiters will be another £100 for 4.