時期 and 機会: timing and opportunities
Distinguish when something happens from an opportunity to do it.
When something happens or fits
時期 is read じき and refers to when something occurs or when it is appropriate to do it. 「引っ越しの時期を考えています」 means that the speaker is considering when to move.
「今は忙しい時期です」 says that this is a busy period. It characterizes a span of time rather than a single opportunity.
A chance to act
機会 is read きかい and refers to an opportunity for an action or experience. 「日本語を話す機会が増えました」 means that opportunities to speak Japanese have increased.
「機会があれば参加したいです」 expresses a conditional wish to participate if an opportunity arises. It neither sets a date nor commits to attending.
Suitable timing and an available chance
「時期はよくありませんが、よい機会だと思います」 says that the timing is not good, but the speaker considers it a good opportunity. Timing and the value of the chance can be evaluated separately.
「この時期」 refers to this period, while 「この機会」 refers to this opportunity. Even when a translation uses a broad word such as 'time,' distinguish a time span from a chance to act.
Does 「会う機会がありませんでした」 mean that the speaker forgot the meeting date?
Show the answer and why
No. It means that there was no opportunity to meet. It does not give forgetting a date or not wanting to meet as a reason.