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You can delete the log files directly using FTP, SFTP, your hosting control panel's file manager, or even a script which runs periodically.

For example, if you want to delete all log files which are older than 30 days and your backup output directory is /home/myuser/backups you need to create a CRON job with the command 

find /home/myuser/backups -Btime +30d -name "akeeba.*.log*" -delete

You can run this CRON job every week or so, depending on how often you want to remove log file.

Moreover, keep in mind that log files are useful primarily for troubleshooting. If you have a stable backup you can go to your backup profile's settings and set the Log Level to Errors And Warnings. This will essentially create an empty file unless there are errors and/or warnings during the backup, alleviating the problem.

The Manage Backups page does not check if the log file exists. It always shows a View Log button. When you click that button and if the log file does not exist you are told as much. This will not change.

We don't check for the presence of log files for performance reasons. Log files can have one of two different file name formats with two different extensions and they can be present either in the current output directory or the directory recorded at backup time in the backup record. This means that we'd have to check for the existence of 2 x 2 x 2 = 8 different files for each backup record. Displaying the default 20 records would need to check for the existence of 160 files. This starts getting very slow, especially on the more inexpensive servers.

I forgot one more thing. There is an option in each backup profile, under the quota settings, to limit the number of obsolete records. When an obsolete record is automatically deleted, so does its log file.

I am not sure if that's something that would work in your use case. If I understand correctly you want to remove the log files but keep the database records, in which case this feature does not help you.