As Robinson prepares to spearhead the Sussex attack in their County Championship opener against Lancashire, starting at Old Trafford on Thursday, he has in mind the importance of a good season kick start for his prospects of earning a maiden Test cap this summer.
Having spent time on the fringes of Test selection as part of numerous bio-secure squad bubbles over the past year, Robinson revealed last week that he had been advised by England to aim towards selection for the two Tests against New Zealand in June.
Gillespie, the former Australia quick who played 71 Tests, coached Robinson initially at Yorkshire and more recently at Sussex, and he believes Robinson is ready for higher honours.
"He wouldn't let anyone down. If he was thrown the ball in a Test match, he wouldn't let England down, he would perform a role very well," Gillespie told ESPNcricinfo from Adelaide, where he is now head coach of South Australia and Adelaide Strikers. "He's one of the most researched bowlers that I've worked with.
"What I mean by that is he comes very prepared to every session, every team meeting, he does his research on the opposition batsmen, he watches a lot of footage, he pores over statistics, which may surprise a few people. And I'll be honest with you, I didn't necessarily see that in the short time Ollie was at Yorkshire, but once I got to Sussex I'd seen a lot of growth in Ollie as a player and person."
Having reached this point on the back of two hugely successful Championship seasons in 2018 and 2019 and a match-winning contribution against Australia A on England Lions' tour of Australia early last year, Robinson knows first-hand how crucial the coming weeks can be.
Could Robinson feature during this winter's Ashes series? Gillespie believes he should.
"I think he's good enough," Gillespie said. "I certainly believe he could perform a role for England. I think he's certainly someone England should have in their squad for the Ashes because I think he could have an impact.
"England have got some fine bowlers, there's no doubt about that, and I think he deserves to be in those conversations to be a part of that tour and a part of that squad. Speaking to Australian batsmen who played in that A series, they were very impressed with him."
It wasn't always this way with Robinson. Some observers may have been surprised by his progress, namely those who remember him as a youngster at Yorkshire with loads of talent but less aptitude for discipline, which ultimately saw him released in 2014 at the age of just 20 for a series of misdemeanours relating to punctuality, training and commitment.
Gillespie, Yorkshire's coach at the time, believes Robinson's problems stemmed from a struggle to settle after moving north from his native Kent.
The stepson of Paul Farbrace, who was then Yorkshire's second-team coach before going on to become England assistant coach and now Warwickshire's director of cricket, Robinson missed his friends, according to Gillespie, who remembers Robinson driving back and forth four-and-a-half hours each way in an unsustainable attempt to maintain his social life back in Kent. Robinson's cricket suffered - he would be late for training or not turn up at all and Yorkshire lost patience.
But by the time the pair were reunited at Sussex - when Gillespie took over as coach in 2018 - Robinson was thriving, having adopted a truly professional approach to his game.
"It's fair to say Ollie, being from Kent, he didn't quite settle into the environment at Yorkshire," Gillespie says. "He was quite young and he missed his life down south so it just didn't quite work out. Everyone saw that the skills and the talent were there.
"He's a real leader around Sussex. That's probably something that I think may have surprised a lot of people at Yorkshire.
"Full credit to him. Ollie being released by Yorkshire, he had a choice to make - he could either mope around and be disappointed about it and 'woe is me' or do something about it.
"He sought opportunity elsewhere, he did the hard yards, he got offered an opportunity to trial at Sussex and he did everything he could to make the most of that opportunity. He deserves a lot of respect around the county circuit for how he's gone about things."
Valkerie Baynes is a general editor at ESPNcricinfo