I mean
1962 - finished 3rd, beat Montreal (who finished 1st), lost to Toronto (who finished 2nd)
1963 - finished 2nd, lost to Detroit (who finished 4th), the top four were really bunched together in standings
1964 - finished 2nd, lost to Detroit (who finished 4th)
1965 - finished 3rd, beat Detroit (who finished 1st), lost to Montreal (who finished 2nd)
1966 - finished 2nd, lost to Detroit (who finished 4th)
1967 - finished 1st, lost to Toronto (who finished 3rd)
1968 - finished 4th, beat New York (who finished 2nd), lost to Montreal (who finished 1st)
1969 - finished last, no playoffs
1970 - finished 1st, beat Detroit (who finished 3rd), lost to Boston (who finished 2nd)
1971 - switched divisions and finished 1st against former expansion teams, beat Philadelphia and New York (who finished 2nd in East), lost to Montreal (who finished 1st in East)
Add in 1961 itself where they finished 3rd and beat Montreal (who finished 1st) and Detroit (who finished 4th)
There are really only a couple seasons that stand out as big individually isolated disappointments. On the whole, of that 1959-1973 stretch you'd have probably liked.. 2, 3 Cups from that era based on their regular seasons, taking into account they essentially got one they shouldn't have. So it's a bit disappointing but it's not crazy out of line with regular seasons overall.
In addition to their playoff failures, Chicago also had a reputation for choking when it came to winning the league in the regular season, which meant more back then than it does now. In those seasons when they finished 2nd or 3rd, they were frequently in 1st place for much of the season and then lost it in the final month or so.
So I'm not sure it's accurate to say they weren't really expected to win the Cup because they didn't win the league. There were several seasons where they were expected to win the league for much of the season, and no doubt expectations were high for the Cup as well.
1962-63
Late collapse. As of February 24, 1963, Chicago was in first place with a 30-15-14 record and 74 points. Montreal was second with 66 points, and Toronto third with 65. Then the Hawks finished with only 2 wins in their final 11 games, including 2 losses to Toronto who beat them by 1 point in the final standings.
Bobby Hull missed 4 of the final 11 games and only scored 1 goal in 0 assists in the 7 games he played. Glenn Hall allowed 35 goals in 10 games.
1963-64
They were close all season with Montreal and lost by 1 point. No collapse.
1964-65
Another late collapse. As of February 24, 1965, the Hawks were 32-20-6. They were in 1st place, 3 points ahead of Montreal and 8 points ahead of Detroit and Toronto. Although they had played 2-3 games more than the others, they were also first in W% (0.603 to Montreal's 0.598). Then they only won 2 of their final 12 games, including 5 straight losses to finish the season. Detroit won 12 of their final 15 and finished 11 points ahead of Chicago. Montreal was just 7-5-2 to finish but that still put them 7 points ahead of Chicago in the end.
Bobby Hull scored only 1 goal and 4 points in the final 10 games. The whole team scored only 7 goals in those last 5 losses.
1965-66
Yet another late collapse. Halfway through the season, on January 13, 1966, the Hawks were 21-10-4 and led Detroit and Montreal by 4 points each. On Feb 16, they were 28-16-7 and still led Montreal and Detroit by 4 points (Montreal had 2 games in hand). Through March 2, they were 32-18-7 and led Montreal by 1 point, and Montreal had a game in hand so they were essentially tied. And then, once again, they lost it down the stretch, going 5-7-1 while Montreal went 10-4-0 and easily won the league.
Some said the team wore themselves out trying to get Bobby Hull to 50 goals. Hull scored his 50th goal on March 2 when they were leading the league.
1966-67
The Hawks finally won the league for the first time in club history. They went wire to wire and closed strong, going 8-4-3 in their final 15, and finishing 17 points clear of second place.
1967-68
A second half collapse. After a terrible 0-6 start, the Hawks went 20-4-9 in their next 33 games and were in first place by 3 points on January 7, 1968. Then they went 12-16-4 to finish the season and dropped to 4th.
1968-69
Another second half collapse. On January 8, 1968 they were 22-14-3 and in 3rd place, only 4 points behind Montreal for first. Then they went 12-19-6 in the remaining games and dropped all the way to last place in the East division among the Original Six teams.
1969-70
Success again. After losing their first 5 games, they kicked it into high gear for the rest of the season and won the league for a second time.
1970-71
Another good season. Finished 3rd in the league. Nobody was catching the Bruins but the Rangers overtook them for 2nd with 5 games to go.
To summarize, Chicago was leading the league late in the season in 4 of 5 seasons from 62-63 through 66-67, was a very close second in the other season, and only won the league once. And then in 67-68 and 68-69 they looked like contenders to win the league and the Cup through the first half of the season, and then collapsed in the second half of the season.
From 1962-63 through 1970-71, the Hawks looked like contenders at the halfway point of every single season. If you just look at the final regular season finishes and compare to their playoff results, it doesn't really capture the expectations they had built each season, or the disappointing late season finishes.