Yes, the Regimental system is a monumental topic and worthy of a small library in and of itself. I originally went to the UK to join the British Army in the 80s because I thought it had all the things that we lost in the Canadian Army, tradition was a big part. But so to was the chance to see more of the world than Pet and Cyprus. I certainly wasn't disappointed. The Regiment can be quite a closed world, and I never really thought I joined the British Army so much as I joined the Gordon Highlanders. The battalion was our world. It was generally recruited in a small geographical area, Aberdeen & NE Scotland, but there were small numbers from elsewhere (myself and Oz Macdonald being the only two non-UK*). That said, I also served in the RE and I found Corps pride, and pride in our wider traditions very real. It used to be that you served in a battalion throughout your career (usually even in multi-bn regiments: RRF, Anglians, etc) but now trickle posting (as per the Corps) is increasingly common in the new amalgamated large regiments.
A good Regiment will instill both pride and high operational standards. Leadership, as in every aspect of military life, is a key driver for this. Although a career will be with the Regiment for life, exposure to the wider Army is common as JNCOs/SNCOs/Officers will regularly be posted out to training or other operational units and thereby will be exposed to the 'gold standard' of the Army.
Lastly, in the space of my 4 years in the battalion, the Gordons undertook 2 summers of Public Duties (Balmoral and Edinburgh), numerous 'Keep the Army in the Public Eye' PR tours in the Regimental Area and had full guard mounting and other duties in garrison. Lots of spit and polish; a bit of a pain but I actually liked it - tourists in Edinburgh couldn't get enough of us ;-) I saw the RCRs and LdSH guard mounting in London, I spoke to one guy and they were having a blast.
But in that same period of time, a platoon was detached to South Armagh, we deployed as the Resident Infantry Coy in the Falklands (Coy+), got smashed on Otterburn and the Highlands more times than I care to remember, and then did an op tour in Belfast in 1990. 'Drillers' may not be 'killers', but a soldier that looks like a shower is not necessarily more operationally effective. The acme of a good soldier is the ability to go between operational/ceremonial duties and stay effective, switched on and with their sense of humour intact.
*I joined the Army in 1987. There had been a Canadian in the Gordons a little before me - I found out the hard way because this guy went AWOL and many guys jokingly wondered if I was going to do the same.