Why not go for App -> AWS Firehose -> S3 architecture. Firehose will automatically take care of batching logs and putting them together. Also it can create partitions. No need to use Lambda at all.
No bespoke logging solution is needed.
I run some Lambda systems at scale and CloudWatch Logs has worked excellent for me. It was completely painless and CloudWatch logs has a lot of tooling that comes with it (backups, streaming, indexing with Elastic Search, etc).
What part of it has not worked well for you?
And searching is very limited. Yes, you can stream them to ElasticSearch => LogStash/Kibana, etc but that's extra steps, and I can do similar things completely without CloudWatch Logs, e.g. Sentry and SumoLogic integrate fine with Lambdas also.
I'm not hating on CloudWatch Logs, and alarms are good. It's just not a fully featured logging solution, and it sounds like you agree with that?
Search is powerful enough to do something like say, "get me all messages with this trace GUID" and S3 exports can be automated[1] and you could probably use the "task-name" to associate with your ticketing system.
[1] http://docs.aws.amazon.com/AmazonCloudWatch/latest/logs/S3Ex...
I was considering Kinesis but my use case (multiple small apps not running on AWS sending medium-sized messages a few times per user session) values simple components and keeping the long-term maintenance effort to the minimum.