The bill in question was known as the "Bayh–Celler amendment". The full text of the bill can be found here (section 3.4, page 143).
One way to answer "how close" the US was to Electoral College reform is to compare to other similar events. It seems that at that time this bill was the "closest" in relative terms the US has ever come to modifying the Electoral College system. Passage by the House represented more support than any other proposal had ever received.
But that only measures success of the bill in a one-dimensional way, when by design amendment of the US Constitution requires a complex set of independent political components to all provide sufficient support. So to better understand the context of the bill at the time, it seems worth breaking down where there was support, opposition, and what opportunities did / did not exist at the time.
The distinct political components were: support of Congress, of the President, and of the States; all of which would be required to successfully amend the constitution. I've organized the remaining material along these lines.
In the House of Representatives the bill was passed with fairly overwhelming support in a vote of 339 to 70 (ref). This easily exceeded the 2/3rds vote required for an amendment in that body (ref).
As far as I can tell, the success in the House was the furthest any attempt at reform has ever gotten, although there have been other attempts (based on ref and ref).
As noted in the question, the bill failed in the Senate due to a filibuster. The filibuster procedure at the time required a 60% majority to progress any legislation, which clearly it did not have. Apparently the bill would have gotten 55-60 Senate votes were it put up for a full vote (NY Times 1 and 2), but this still would have been well short of the 2/3rds requirement for an amendment bill to succeed in the Senate. (By Sept. 1970, support had even diminished slightly, as evidenced by a second failed cloture vote).
On the other hand, those Senators opposed may have felt there was some risk, or else the filibuster would not have seemed necessary. From the above article which was reported prior to the filibuster:
Senator Bayh said today it would be “a really tough job” to get the
amendment ratified in time to make it effective for the 1972
Presidential election, but he thought there was “a good fighting
chance” to do so.
So Bayh thought there was not only the possibility of enough votes, but also that the process could occur within roughly a 2.5 year period. This might have been unrealistic optimism, or perhaps the main opposition really was just those senators who ultimately decided to filibuster and not risk a full vote.
One reason for the focus on the Electoral College at that time was the prior presidential election of 1968. In that election Richard Nixon won the electoral vote by 56% to 36% of his closest rival, but despite the seemingly comfortable victory this actually represented a very close popular win of just 43.4% to 42.7% (ref).
President Nixon publicly supported this type of reform and had made statements to that effect on more than one occasion. In his February 20, 1969 Special Message to the Congress on Electoral Reform, Nixon wrote,
Today, our presidential selection mechanism ... requires overhaul to
repair defects spotlighted by the circumstances of 1968. The reforms
that I propose are basic in need and desirability. They are changes
which I believe should be given the earliest attention by the
Congress. I have not abandoned my personal feeling, stated in October
and November 1968, that the candidate who wins the most popular votes
should become President.
...because of the compelling specific weaknesses focused in 1968, I am
urging Congress to concentrate its attention on formulating a system
that can receive the requisite Congressional and State approval.
Different plans for reform have been responsibly advanced by Members
of Congress and distinguished private groups and individuals. These
plans have my respect and they merit serious consideration by the
Congress. I have in the past supported the proportional plan of
electoral reform... But I am not wedded to the details of this plan or
any other specific plan. I will support any plan that moves toward the
following objectives: first, the abolition of individual electors;
second, allocation to Presidential candidates of the electoral vote of
each State and the District of Columbia in a manner that may more
closely approximate the popular vote...
Taking him at his words here, it seems clear that he would have supported the amendment bill. [As noted in a comment from T.E.D.] although the president does not formally approve / veto an amendment proposal, he could have exerted political pressure to aid its passage. For instance, by tying his support for other political activities to the support of others for the amendment.
In the case of the Senators opposing the 1970 proposal, they were from both parties; perhaps Nixon could have been more effective at mustering influence on those from his own party and not all those who eventually filibustered. However by Sept., 1970 apparently he had not done so (ref).
Support of the States
In "A Survey Finds 30 Legislatures Favor Direct Vote For President", The New York Times, October 8, 1969, it was reported that:
30 state legislatures were "either certain or likely to approve a
constitutional amendment embodying the direct election plan if it
passes its final Congressional test in the Senate." Ratification of 38
state legislatures would have been needed for adoption. The paper also
reported that six other states had yet to state a preference, six were
leaning toward opposition, and eight were solidly opposed.
Surveys can have a degree of error, of course, so its worth trying to weigh its results against other information.
In the failed Senate vote, the Senators in opposition "were evenly divided politically, 18 Democrats and 18 Republicans. It was almost entirely a coalition of Southerners and Conservatives from small states who had protested that abolishing the Electoral College would reduce their states political influence." (ref) Possibly or likely, the same set of states were among those clearly not in support of the amendment.
It is of course unknown how success in Congress might have influenced the states to behave; possibly such hypothetical support could have nudged enough states over the line. The strong support in the House of Representatives appears to be greater than the surveyed support of the state legislatures, which might indicate that general popular support was also strong. Such support could have changed enough minds (or at least votes) in enough states.
But regardless, the support of the states at that particular moment was not nearly as important as Presidential & Congressional support, because the amendment bill specified no expiration before ratification was required. Therefore it would have been possible for state ratifications to gradually accrue (as later happened with the 27th amendment which took about 200 years to ratify.)
Given this last fact, and accepting the survey as being at least in the ballpark, eventually sufficient state ratifications would have been quite likely if not actually inevitable.
So, "how close" was the US to electoral reform in 1970?
I'd say, "pretty close", in the sense that this was not a fringe idea or one lacking substance. From what I gather, it was essentially only the Senate filibuster that stood in the way, because all other dependencies were either already satisfied or had a path forward.
Also in hindsight it seems quite significant that the US did not ultimately make any changes in 1970, because two elections since then (2000 and 2016) would have seen different winners had the proposed changes been in place.
Note on terminology - a proposed amendment as it makes its way through Congress is not really a "bill" in the sense of "how a bill becomes a law" because it has a different standard for passage (2/3rds majority in both houses) and does not require a presidential signature (nor veto). Technically these proposals are called "Joint Resolutions". However other activities in Congress also go by the term "Joint Resolution" and so it doesn't unambiguously refer to proposed amendments. So I'm using the term "bill" here just for simplicity, and because the congressional process is nearly the same.
Note on references - in some cases I've gotten information from Wikipedia; although in these cases Wikipedia itself cites primary or secondary sources, I haven't had access to all of those. I directly linked to those where I could.