It's not really a case of ships having a short design life, the existing T23s are pretty old already. Materially they simply need replacing as they are already beyond their original design life. They are still capable but are beginning to show their age and there's only so many times you can put them through refit before it's more cost effective to replace them.
A new design also benefits from being built around current weapon systems, sensors, propulsion etc rather than trying to retrofit them into an older platform which then results in compromises in their functionality.
As for the statement that the UK doesn't need a big - or more specifically a capable and globally deployable - navy, the simple answer is the UK does:
98% of our trade by bulk travels by sea. Simple economics mean part of the price you pay for anything bought in the UK which has been shipped by sea, including fuel (fuels for cars, electricity bills for gas powered power stations etc) covers transport costs. Those costs include a) the cost of moving it and b) incidentals like insurance. If the SLOCs (sea lines of communication) are not secure then you have the potential that commercial shipping avoids hazardous areas such as the Bab-al-mendeb, Hormuz, Indian Ocean etc meaning they travel further at increased cost and the cost will be passed onto the consumer. Alternatively, if they continue to travel through hazardous areas insurance costs go up and will also be passed onto the consumer. Hence it's in the UK's direct interest in maintaining security of the SLOCs, so a surface presence supports that especially a lightweight frigate designed to operate independently and more focused on what we refer to as 'constabulary operations' rather than high-intensity war-fighting. Going back to energy supplies, I can't remember what the specific details are but there is basically a conveyor-belt of ships carrying LNG and oil to the UK for use in electricity generation; if that supply is disrupted our reserves in the UK allow for power generation for 2-3 weeks max before the lights go out - that is how dependent we are on foreign energy and hence there is a clear interest to the UK in securing that supply.
Additionally, the UK's economy is very much a global one with UK companies having interests in many regions of the world. It is therefore in the interest of UK PLC to help maintain security wherever there are UK interests. There is a large value politically in having a presence in these regions and again, a lightweight frigate which can conduct maritime security, presence and influence operations has a clear utility in this. They show the UK is politically committed to a region and if security does deteriorate have a very practical utility in intelligence gathering and providing a military capability until the cavalry (i.e the carrier strike group) arrives.
Regarding the carrier strike capability, this is not really about placating the Admirals but again about giving the UK the ability to secure our interests at range. Again, our economy and energy supply are dependent on maintaining security. A carrier strike group gives you ability to forcibly enter a theatre, conduct day 1 of the war operations onwards against all but the most hardened of threats to rectify the situation and then disengage without having to rely on host nation support or potentially put boots on the ground (although this is also catered for if required with the UK's amphibious capability). This is HMG's preferred MO and referred to as 'strike warfare' - go in, decapitate the snake and withdraw without getting embroiled in a long-term conflict.
Apologies, rant(?) over.
TL;DR - the average person in the UK takes for granted the global maritime security we protect and is generally ignorant of how dependent we all are on it. That would change soon enough if the price of consumer goods and food go up or the lights go out if the SLOCs are disrupted. To do this you need ships. Expensive ships designed for high intensity war-fighting are great but you can only afford so many, a cheaper, lightweight frigate therefore enables more hulls able to be in more places at once - in this case, quantity really does have a quality of it's own.
Edit: As for migration, yes we do that too. However, in UK waters this is really the responsibility of Border Force. We are more focused on trying to prevent the problem at source (i.e the regions where security is so bad that traveling across Europe and taking to the seas in a tiny boat in the hope of the better life seems worth the risk. Most of these regions are in economic problems because of the underlying security issues.)
