Born Finglas, County Dublin, Republic of Ireland, Braddish initially played junior football with Raven Athletic in his local Finglas area. During the 1970s this club served as a nursery team for Dundalk and in 1975 they recruited Braddish. Braddish subsequently played five seasons with Dundalk under manager Jim McLaughlin, helping the club win seven trophies including the League of Ireland title in 1976 and the FAI Cup in both 1977 and 1981. His team-mates at the club included, among others, Terry Flanagan, Dermot Keely and Cathal Muckian. Described as an elegant and intelligent midfielder, Braddish scored 17 goals during his time at Dundalk – 12 in the league and 7 in cup competitions. He scored his first goal for the club on 17 August 1975 against Drogheda United in a 2–2 draw in the Donegan Cup. Dundalk eventually won the game 3–1 on penalties. On 6 April 1977 at Dalymount Park, Braddish scored in a 1–0 win against St Patrick's Athletic in an FAI Cup semi-final replay. On 23 January 1978 he scored a hat-trick in the league in a 6–2 win against Shelbourne. Braddish was on the verge of signing for Los Angeles Skyhawks when in April 1978 he was offered a contract by Liverpool. Together with Derek Carroll and Brian Duff, he was one of three Dundalk players Liverpool signed for a combined fee of £50,000. However none of the trio managed to get a first team game and all of them subsequently returned to Dundalk. On 29 January 1980 Braddish scored a second hat trick for Dundalk, again against Shelbourne, helping the club to a 9–0 win in the league. After leaving Dundalk, Braddish went on to play for St Patrick's Athletic for a further five seasons. Among his teammates at this club were Dave Henderson and a young Paul McGrath.
Republic of Ireland international
Braddish represented the Republic of Ireland at schoolboy, youth, amateur and under 21 levels. After an impressive performance in an under 21 international against Northern Ireland in March 1978, he was then called up to the senior squad. He made 2 appearances for the Republic of Ireland, winning both caps in April 1978. He made his senior international debut, under Johnny Giles, on 5 April in a 4–2 win against Turkey. He came on as a late substitute in the 67th minute during a friendly at Lansdowne Road. He then won his second and last cap on 12 April when he started for the Republic in a 3–0 away defeat against Poland.
