What are the options for organizing an "intermediate" server to collect incoming mail?

Question

Mail server

The background is as follows: we use Yandex traffic rules, there are a dozen mailboxes, users sit in Thunderbird via IMAP. The problem is that during "Moscow" peak hours, i.e. after about 9-10 am Moscow time, various brakes begin and increase throughout the day when working with incoming (and sometimes when sending) letters - from extremely slowly loading attachments to banal text letters, in which even the body does not load for 10 -20 minutes, and the IMAP server "non-response" errors that have become commonplace. There are definitely no complaints about the width and stability of the Internet channel, the problem is clearly not in it.
Considering the fact that this happens almost daily at the same hours, and also the fact that changing email clients does not help at all, the thought arises that Yandex simply cannot cope with peak loads, and it is by IMAP, because when switching to the Yandex web interface, it removes the brakes as if by hand. But compared to the advanced functionality and flexibility of the Mozilla client, it is very minimalist - in general, employees categorically do not want to use it.
The Mozilla client is configured to store letters locally for 60 days, but often users look for and open older correspondence, sometimes in the archive for previous years - and it is not possible to increase the local storage period in the client to such figures, because the disk space is not very large. then rubber, and Thunderbird does not like large databases and huge search indexes, it starts to slow down with every sneeze. The problem needs to be solved in a different way. For example, your own mail server.
Creating your own fully functional server is a good idea, of course, but there is one thing. I do not want to waste time in the future on fighting spam, getting into emergency situations and other security issues, which now almost entirely fall on the shoulders of Yandex. Therefore, the following question arose - are there any working options for organizing an “intermediate” IMAP mail server in order to collect incoming mail so that during peak hours, at least letters already downloaded from the Yandex server do not have to be opened for 10-20 minutes?

Answer 1

Problem for several years.
1. Yandex provides the service for free, what do you want for free? 2. Yandex does not have support for this service as such, you can contact the support by specifying the ip
address of the office, etc., but the result will be zero
for some reason I want everything to be filtered for you for free
4. where is free cheese usually found?

Answer 2

are there any working options for organizing an "intermediate" IMAP mail server

No. Because there is no need.
Under-offices keep mail in the "clouds" - in Google, Yandex, at the bottom of the bald. This is their decision, they bear all the risks of this decision, although it also has advantages, but usually they do not cover the risks.
Normal offices keep mail on their server, which they control completely. Here, too, of course, there are risks, they must be taken into account.