Would you have Budge or Perry over Connors?
Whilst the indoor surfaces helped Vines, do you not think the pro tour format (loads of matches close together) hurt him? With his reputation as someone who could blow hot and cold in terms of hitting his peak he seems to have been far more suited to tournament play. Of course, he played virtually no tournaments during his best pro years so it's hard to judge.
Is it really true that a tournament format would help someone who tends to run hot and cold? In a H2H series, if you have a bad day, you've just lost one match; you can pick up again the next day where you left off. In a tournament, if you have a bad day, you could be eliminated.
Anyway I think Vines' reputation for running hot and cold can be exaggerated. Studying his record, I've been surprised to find so much consistency.
Tilden said once that when Vines was good, he was invincible, but when he was bad he was very bad. That's not quite the same as saying that he was often bad. I think it just means that when he was bad, he hit extreme lows. Tilden thought that Budge's best, in a single match, was not quite as high as Vines', but he said that when Budge had an off day he was still great. That's why, when asked who was the best player he'd ever seen, he said Vines on a single day, but Budge over 365 days.
I'm not sure if that really answers your question -- because I'm not entirely sure myself, whether Vines was more suited to tours or to tournaments. Maybe it was neither.
I wouldn't place any of Budge, Perry or Vines above Connors. That's not because I think it's a ridiculous position to put their names ahead of Jimmy's -- but I would find it a hard sell. I tend of think of Connors as being in a second tier, with GOAT candidates in the first tier (which I usually think of as Tilden, Pancho, Rosewall, Laver, Borg, Sampras, Federer). Connors, imo, is not far behind that first tier. He's there because he was clearly second-best to Borg; but he had a long career and added many late-career accomplishments that Borg did not. So if Jimmy is second-tier, it's not by a lot.
Now if I compare Budge's career, for example, to Tilden's, I see a large gap. If I compare Vines' career to that of Pancho Gonzalez (and their careers are somewhat similar, which makes it easier to compare), I think Pancho's was clearly superior -- I gave some of my reasoning in my post above where I compared their H2H tour victories. I could throw in Pancho's longevity as well.
Vines just did not dominate the 30s the way Pancho did the 50s. It's true that Vines can be seen as world #1 in many years -- but he was usually co-#1, with Tilden or Perry. He was clearly #1, alone, in 1932. I think he also has a good case to be named #1 in 1931 -- but I doubt he would have defeated Tilden that year in direct competition. I'd also say Vines has a case to be #1 in 1934-35 and 37 (imo his competition in '36 is deeply unimpressive, and inferior to what Perry faced) -- but Perry was sweeping the Grand Slam majors and Davis Cup in those years against fine competition. So who was better? Isn't it reasonable in these years to name them as co-#1's? When they finally met in '37, they split their matches evenly, despite the matches being played mostly indoors, to Vines' advantage.
The 1950s was a totally different case. Pancho defeated his tour opponents by clear margins; and no one in the amateurs was at his level. He was a clear world #1 for a long string of years.
Connors, we usually only give 3 years as world #1 (1974, 1976, 1982). That seems to be well behind Vines, if we give Vines, let's say, 5 years as #1: 1931-32, 1934-35, 1937 (I'm not entirely sold on all those years, but I think this is reasonable list). But that's with co-#1's. Wouldn't Connors have more years as #1 if he played in a pro/am split that was similar to the 1930s?
In 1975, for example, none of us picks Connors as #1. But imagine the tour split among pros and amateurs. Let's say Connors, after his brilliant '74, turns pro. In '75, like Vines, he faces older opponents on the pro circuit and dominates them (let's say, for example, Newcombe and Laver). Meanwhile Ashe and Orantes are sweeping the traditional majors. With the display of power and excellence that Connors put on in '74, he would still be thought of as world #1 in '75. Maybe he'd be named as co-#1 with the best amateurs (Ashe and Orantes), but there would be no way to say that he was #2; because as long as he defeats the pros in front of him, he's not dethroned.
In other words, in a pro/am split like the 1930s, in which no one is sure whether the top amateurs or the top pros are superior, Connors could be seen as world #1 from 1974 perhaps all the way through 1978: five years altogether, equaling Vines.
Didn't mean to make this such a long answer, but in short, Vines/Perry/Budge are in my opinion somewhere in the second tier with Connors; and if forced to choose I'd put Jimmy's name above theirs.