I think a couple of things are getting mixed up here.

For the process of amending the Constitution, as laid out in the 1787 document (Article V), here is the original language:

The Congress, whenever two thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two thirds of the several States, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as Part of this Constitution, when ratified by the Legislatures of three fourths of the several States, or by Conventions in three fourths thereof, as the one or the other Mode of Ratification may be proposed by the Congress; Provided that no Amendment which may be made prior to the Year One thousand eight hundred and eight shall in any Manner affect the first and fourth Clauses in the Ninth Section of the first Article; and that no State, without its Consent, shall be deprived of its equal Suffrage in the Senate.

So what that means: first a Constitutional amendment needs to be passed by two thirds of each house of Congress, or two thirds of the states need to call for a Convention to propose amendments.

However, once an Amendment passes this stage, it needs approval by three fourths of the states. This can either come in the form of the state legislatures voting for the amendment, or through special state conventions elected to vote for or against the amendment.

Every single constitutional amendment to date has been passed through Congress. All of those amendments bar one (the 21st Amendment, repealing the 18th Amendment) were approved by state legislatures, rather than state conventions.

The method of proposing amendments through a Convention requested by at least two thirds of the states has to date never actually been attempted, and it's not very clear just how this process would even work in practice.

The notable feature of Section 6 is less the ratification requirement of three fourths of the states, which is constitutionally-required, but the deadline for ratification of seven years. This is a feature for constitutional amendments that was introduced in the 20th century - the 18th, 20th, 21st and 22nd Amendments have deadlines in the text of the amendment, and the 23rd, 24th, 25th and 26th amendments have deadlines in the introductory text of the amendment as passed by Congress. The 27th Amendment notably had no deadline as it was originally approved by Congress as part of the package of Amendments known as the Bill of Rights, was ratified by a few states, and then largely left "dormant" until being rediscovered by an undergraduate student at University of Texas Austin, Gregory Watson. His paper on the amendment got a C from his TA, but ultimately his research led to a campaign to get the amendment ratified by three fourths of the states, which happened in 1992.

So why time limits? Richard F. Hamm's Shaping the Eighteenth Amendment gives some clues as to why this particular amendment was the first one to have such a rider successfully attached. Opponents of Prohibition had originally wanted a two year grace period for implementation after ratification, and/or compensation to brewers and distilleries - Congress eventually came up with a one year period. Opponents also saw a seven year window for ratification as "fair" - the original proposal from then Ohio Senator Warren Harding was for four years. It was basically a means of weakening attempts to block the passage in Congress by providing a window of time to prevent passage in the states.

Of course, it's not super clear just how constitutional or binding time limits are. In 1921 the Supreme Court heard Dillon v. Gloss, in which, to make a long story short, an arrest under the Volstead Act (enforcing alcohol prohibition) was contested under the pretext that adding a time limit to the 18th Amendment invalidated the whole amendment process. The Supreme Court basically said: "Time limits are fine, we guess." But notably the 18th Amendment had been ratified by the necessary states before the time limit was up, so it wasn't necessarily a very tough call to make.

Of course there is a potentially interesting case to be made about the legality of time limits because of the Equal Rights Amendment, passed in Congress in 1972 with a seven year limit, that in 1978 was extended to a ten year limit (ie, to 1982). This ten year limit was set by a resolution voted on by a simple majority, and in any case due to increasing political resistance, it was not passed by the necessary states by 1982 (although four states rescinded their ratifications in the 1970s, and this isn't something provided for or forbidden in the Constitution). Of course several states have since ratified the amendment, with Illinois being the 37th in 2017. If a 38th state ratified the amendment - well, it's not 100% clear just what would happen.

To go back to Article V, note the two exceptions to the amendment process: no amendment could ban the slave trade before 1808 (when Congress was constitutionally allowed to ban it), and no amendment can change the number of Senators each states has (it's theoretically an un-amendable part of the Constitution, unless each State agrees to it).