By a previous Lib Dem/Ind admin, following support from the broader town population in a consultation. It was reversed very quickly by incoming Tories, the very first thing they did, after business owners started complaining vociferously that they could no longer park their personal cars immediately outside their front doors, sorry I mean that their customers (the vast majority on foot) couldn't access their stores. The problem with closing Torbay Road and the level crossing to traffic is there are few suitable alternative roads crossing the railway through the town.
I agree about underpasses generally but this site might be a challenge as it's rather low. The entire area between railway and current sea wall is reclaimed salt marsh. The railway embankment rises a little above that, and sea defences are planned to be improved over the next few years. The Netherlands manages to construct underpasses all over the place of course, in a country where much of the land is below sea level! If a wide underpass was practical at Paignton, the south side of it might be fenced off as 'railside' with separate platform access ramps, a way to achieve step-free access to platform #1 from the booking office side without leaving the station. That could allow Paignton to become ticket-gated with only one barrier line on the town side.
The big advantage of going under rather than over is you only have to clear a human or cyclist rather than the tallest rail vehicle plus electrification clearances etc, so ramps can be significantly shorter. The UK seems to have a particular aversion to underpasses, no doubt partly due to some very poorly designed implementations from the 1960s and 70s, and lack of subsequent maintenance and security monitoring. In many town centre developments today, examples from this period are being removed. Making approach ramps broad and open, ensuring there aren't any hidden areas etc, can help design out antisocial potential.
Another possibility might be that the crossing could be closed to normal traffic, but full vehicle access to the road might be retained as a cul-de-sac from the east (the sea front side). That would divorce the level crossing issue from the business premises access issue, though they'd still no doubt start moaning about loss of 'passing trade'.