"Do agents actually look after any problems with the property that might and probably will arise"
I have let a flat on what is termed full management and have used letting agents for ten years, these are my observations.
Full management is somewhat of a misnomer, what you actually pay for, which incidentally costs me 11 and half per cent, plus VAT of the monthly rent, is Monday to Friday management and that's pretty standard whoever you might appoint. If the tenant has a problem over the weekend they are on their own from 5 pm on Friday to 9 am on Monday, there is no emergency number supplied. On one occasion my tenants had reason to call out an emergency plumber over that period and they tend to be more costly. I would also advise that I have found that the letting agents panel of appointed tradesmen are top dollar and not particularly the best, for example, we had one firm of plumbers that had to redo some work carried out, three months down the line, for which they charged again. In the end I refused to use them and they, the agents had to find an alternative. There have been occasions when I felt it would have been more expedient if I'd sorted the problem out myself, but then that's what the landlord pays a third party for. All in all, you pay a lot for very little, an annual inspection plus all the necessary checks to make sure your property is fully compliant and safe to let, again not unreasonably you do pay for all those checks at the outset. However, I do sometimes feel that it would be easier for us to have sourced our own plumbers and electricians etc. because at times I think we have been ripped off by the ones the letting agents used.
Also be aware that legislation is changing regarding no fault evictions, something that has had repercussions for newly resigned homelessness minister no! you couldn't make it up even if she did actually want to sell the property, the tenants I believe were evicted, which isn't good optics for someone who is a homeless minister, and when the property didn't sell, she re-let it at a vastly hiked monthly rent. None of that is illegal at the moment, but will change.
I would also advise that you take out an insurance policy to protect you if the tenants default on their rent, the insurers will reimburse you for those lost payments and they would also pay any court costs that would arise in prospective evictions if you have the right grounds for that. Your letting agent should also advise you on the best policy, usually you can do that through them.
However, if you get all your ducks in a row, quite possibly using a letting agent is the best option because many landlords don't want to be hands on, especially if you live some distance from your property. I wish you good luck with that, it is doable but be aware of the pitfalls.