As well as the reasons Sam mentions, the length-declaration stage serves another purpose: it's the contestants' only opportunity to declare they haven't written their word down. Even if you were unconvinced that the tactical advantage of declaring second improved the game, you couldn't get rid of the declaration stage without also changing the rules in some other way to cope with not-written-down words.
However, Countdown didn't require you to write anything down until about the 1990s(?) - if you had the same word as your opponent it was simply taken on trust, so it appears that wasn't why the length-declaration stage was invented. If you go further back to Des Chiffres et des Lettres before Countdown started, they didn't have to deal with matching declarations because the rule then was that in a tied round only the picker scored. The picker would declare their length first, and the other player would only declare if they had a longer word, otherwise they'd reply "pas mieux" (not better) and we'd never find out their word or length. See
DCedL's first episode from 1972 and a later episode from
well into the 1980s after Countdown had been running for a few years.
If length-declaration was introduced for the tactical reasons, that was some well-placed ingenuity. Suppose the format didn't exist and you were inventing it today - having come up with a game where the contestants had to find the longest word they could from a selection of letters, the subtle tactics introduced by having both contestants declare their lengths first rather than just offering their words wouldn't necessarily occur to you. Perhaps it occurred to Armand Jammot, though.
Since
episode 1, Countdown has always given both players the points if the round is tied. This seems to have been a decision made quite late in Countdown's development. The
Calendar Countdown pilot and
actual Calendar Countdown, for example, copied the French rule, crediting only the picker when a round was tied.
Between Calendar Countdown and Countdown, they must have changed this and decided to award points to both players in a tied round, but the mechanics of asking for the length and then the word still remained. At this point, was length-declaration no more than a useless fossil of the show's French ancestor? Perhaps they wanted to save time by not asking for a beaten word? It would seem strange to make a system which would regularly have to ask each contestant two questions in the name of saving time, though.
The early producers of Countdown could have removed the length-declaration stage at any time if they wanted to, until the not-written-down rule came in. This seems to lead to the inescapable conclusion that they recognised that it subtly changed the tactics of the game and deliberately kept it for that reason.