Consider: What sort of terrain will I use it on? Do I live in Cambridgeshire (flat), or Plymouth (hilly)? The 1x revolution in gravel bikes (made for hills, the traditional reason for more gears) suggests that bikes have long been encumbered by too many gears, but often of insufficient range at the lower end.
Overall, you got a nice bike, with a very nice butted steel frame likely to last you till you sell, give it to a friend, or it gets nicked.
If the groupset isn't worn out, use it till it is. 105's fine. Ultegra's better, and SRAM's Double-Tap (D.T. - see what they did there?) is a work of intelligent simplification - but they're expensive, and finding vintage parts or figuring out how to fit new parts to vintage bikes is a bit of a hobby that will eat into your time on the saddle (writing as a seasoned bike-fiddler with a burgeoning tummy).
I live in Hong Kong, which is about as hilly as it gets. I have an old Merida road bike, 105, with a much more primitive frame than yours. It's pretty heavy. I took off the 52T ring and run it 1x7, and still only use the fastest gear for a few minutes at a time, racing through evening traffic. One day things will start to wear out, and I'll look into lower gearing in the form of a smaller ring yet, or larger sprockets. Do you really ride that fast? How often do you actually use the highest gear, and for how long?
Down-tube shifters take getting used to, but it's not insurmountable. Bar-ends are a good alternative, but my next 'project' will be an experimental remounting of the remaining shifter onto the top tube (keen to avoid more cables poking from the bars). More simply, you can hack your down-tube shifters with an extension (plunge-pole unscrewed from a shattered French press coffee-pot lashed on with Sellotape, in my case, but half a stout chopstick would do just as well).
Indexing requires more adjustment, so I turned it off. One less job to do at home.
These homebrew aesthetics may not instantly appeal to you, and your local bike shop will hate you, but you've spent a bit of cash on a good frame (the best part to have spent it on), so you may wish to consider the other end of the spectrum of options you have - hacks, sidegrades, and upgrading your skills. Down-tube shifters force you to shift your position every time you use them (reducing aches on longer rides), to read the road ahead and learn to shift anticipatitively rather than reactively, and to develop your strength and flexibility to feel at home across a wider range of cadences. Ride it for a bit. Then ask yourself very honestly what works pretty well or well enough, and what about it you really wish could be easier/ more comfortable/ more efficient/ more practical/ cooler/ flat-out faster, and so forth.