I confess I do not follow British weapons all that well, but Wikipedia is telling me that Sea Cat was the very first point defense AA missile system. How it is then obsolete upon installation is a head scratcher for sure. That it may not have worked all that well I can believe. Pioneering systems frequently are not optimal.
I am failing to see how needing to install both the weapon and the launcher is in any way cheaper, which is what you have claimed for the Type 31. Another head scratcher. In these times of networked fleets, it is going to be common for one ship to physically fire another ship's weapons. So, a Type 26 may well load a few AA missiles, and be the nearest platform with the necessary weapons aboard in a situation, thus being selected by the air defense ship to launch. Ships with VLS will carry "Lord knows." Makes the enemy's job harder. That ASW ship that you were ignoring because "it's not an AA ship" suddenly up and turns out to be one after all. Networking will allow this type of flexibility, and the push is on to make more ships fit this sort of system. A Type 26 with a MK 41 VLS will be far more useful beyond just its as-designed role. The Type 31, lacking this sort of launcher system is relegated to only the missions for which it was fitted out. A Type 31 will need to have a MK 41 VLS installed, as well as the weapons that will go in it. Type 26 has the installation already. The money is already spent. Now, it just needs the missiles. In no way comparable. Better than a Type 21? Well, hopefully technologies have improved over time, and we at least aren't going backwards.
The comparison of the Type 21 and Type 31 has some problems:
Sea Cat was a completely obsolete point-defence system at the time the Type 21 was new, i.e. for its time the air defence capabilities of the Type 21 were extremely poor. The same is true for their radars. In addition, with all the mentioned added weapons, the hull was actually overloaded. Limitations regarding the use had to be introduced and still massive cracks were common. No upgrades were possible. The armament had to be reduced to be able to continue to operate the ships. The Type 21 was one of the worst designs of its period - and there were many bad designs.
In contrast, Sea Captor is no point defence system, but can be used to defend also other ships. It is an up-to-date air defence system. The air defence capabilities of a Type 31 will be good (for a GP or ASW frigate, for sure not in comparison with a dedicated air defence ship), way superior to a LCS both in regard of the missiles and the radars.
A Type 26 may have the space to load longer range AAW missiles, but for which of those it has a suitable radar set? The Type 26 is optimised for ASW, it does not have the radars for a optimised air defence ship. For sure, the VLS launcher can be loaded with future weapons, but they have to be bought increasing the coast. In case of a Type 31, also the launchers have to be bought, but it would be still cheaper than a Type 26.
As you have written, the Type 31 is designed to be cheaper and it does include fewer systems, but when introduced the armament will be comparable to a Type 26 (based on what I know regarding what the RN actually has to arm a Type 26) and it still has a lot of capacity for upgrades. It is the much more sensible approach compared to the Type 21, which were designed to be cheap, but were more expensive than aimed for and had no capacity for upgrades.