Neither of those players were "well past their prime" when they joined Spurs.
Ginola became PFA Players' Player of the Year and FWA Footballer of the Year while at Spurs, becoming the first player in Premiership history to win the award while at a club who finished the season outside of the top four.
And although Klinsmann's career was on decline he was still close to his prime - which is why Bayern bought him, and where he scored 31 goals over 2 seasons in the Bundesliga.
Lineker scored 67 goals in 105 league games club.
Top scorer in the First division in 1989-90 season, scoring 24 goals
Second-highest goal scorer in 1991-92 season, scoring 28 goals from 35 games
Won the FA cup in 1991.
But I see what you mean but it's a risk, since not all "fading stars" keep shining once they get a new pay-day secured. Of course, players like Davids and van de Vaart certainly did contribute...