The sawtoothing thing is probably due to the behaviour of some TCP implementation if that is being used in that particular speed test. If a TCP ACK gets lost or intentionally dropped or is very much delayed then the original data packet sender end TCP software will reduce speed a lot and then recover its courage later and this is the speed sawtooth. It is annoying but just quite natural. It results from the sender overdriving the link, going too fast, until intermediate router starts being forced to drop packets and then this causes the problem. A better-designed TCP implementation will find the right speed and send just very slightly under the maximum rate so that things will remain stable.
I would go so far as to say that this is often a bogus phenomenon that just says more about the design of the speed tester or its underlying o/s than your link so it is not testing your link at all but the combination of link plus software which is possibly totally irrelevant to your particular situation with some server x that has whatever software in it. I don’t think speed testers should be using TCP at all, not if the aim is to measure the link and nothing else.
Slow initial growth in transfer rate is another bogus TCP behaviour thing related to design of some TCP implementation again, plus various other environmental factors.
I do not have much respect for most speed testers as they are apparently written by morons who do not understand what they should be aiming for and are lazily just using the o/s normal facilities which are not suitable as they have totally incompatible goals that override and mess up the results.
If I ever design one it will carefully check that the link is quiet in terms of alien traffic and then slowly increase test packet send rate with carefully controlled per-packet timing, monitoring transit time all the while, so as to detect the point at which overload begins. That way the link speed can be established properly and in a way that does not depend on random o/s-internal software design decisions.