Does the quality of my instrument effect the likelihood of me getting performance opportunities?
In a word, no. The only real thing that will effect the likelihood of getting gigs is your ability. A better instrument will obviously feel easier to play and possibly boost confidence and may help in the long run but the only thing stopping you getting gigs in the first place will be you. The legendary saxophonist Charlie Parker used to borrow a different horn before each gig (mainly because after each gig he’d hoc the current sax for heroin!). Many great players start gigging on ‘student’ instrument, only upgrading once they can justify the cost based on income from work.
However, some investment can be wise. Guitarists and bass players may want to consider their setup – e.g. amps and pedals – this can help you diversify for different sounding groups. Horn players and singers should own their own mic and stand. Keyboard players should have their own amp and, if the intention is to get into pop music or session playing, be able to program DAWs and keyboards as they’ll often lead the triggers.