Reddit mentions of Selections Kingfisher Hollow Tine Lawn Aerator

Hand weeding and reseeding is fine just be sure to add some soil to fill up the holes created when you weed. I think in the UK you can get away with seeding in spring as you don't get the brutal summer temps that the US does. I reseeded here in Auckland NZ in late spring and we have similar peak summer highs to most of the UK and didn't get any die off. Seed is also cheap, particularly for a small yard, so if you fail no problem just over seed again in Autumn. If it was my lawn I would spot spray the dandelions using glyphosate , wait a week dig them out and then aerate with something like this, add a thin layer of compost (don't smother the existing grass) and then overseed and add a starter fertiliser. Keep the soil moist while the grass is starting out, watering at least once a day depending on the weather. I've never dealt with moss in my lawn but you may need to add lime to change the PH or improve drainage.