Very complicated question:
firstly, there no archaic/modern distinction within these language: both have modern literary tradition. But they can keep archaic features of sub-levels or even proto-conditions;
secondly, you have mistake with 'root word' because savaş/sogıs are derivatives from •√saba-, where -ş/-s are verbal noun formants; and savaşiyor/sogısıp catır(jatır) are derivatives from √savaş-(mak)/√sogıs-(ıw), where -ş-/-s- is the reciprocal;
thirdly, Oghuz formant -Iyor isn't from √cat-(jat-) but from formerly verb √yör(ü)-//to run [it is cognate to cür-(jür-)];
fourthly, it need to clarify that these auxiliary verbs in this case are parts of the present tense - so-called new praesens - and they differentiate in their behaviour from the very same auxiliary verbs when they used for showing a mood or a modality (so another verbs used too): e.g. new praesens is sogısıp cür but as an auxiliary verb is sogısıp cüredi in the present or sogısıp cürdi in the past [there is -A tur- form additionally to the -Ip Aux.Verb form but only with √tur-]; there no such feature in the Oghuz branch except few languages/dialects that was contacted with Kipchak so there only -Iyor is new praesens [ similar but different situation within few Kipchak languages itself; there no so-called new praesens at all in some Kipchak languages, there is only simple present -A-/-y- ];
fifthly, -Iyor is grammaticalized -A yör- and it was corrupted by languages and dialects to even -(I)(r);
sixthly, there was -Ar formant - so-called aorist or present-future - in Kipchak and Oghuz branches, and for Kipchak branch it has tendency to determine the future with sporadically open/close vowels [i.e. -Vr] in it by different Kipchak languages/dialects; in the Oghuz branch it still an aorist but in some Oghuz language/dialects there happened the contamination with new praesens especially where it was corrupted to the -Ir than there is the tendency to indicate the present with -Ir formant and the future with -Ar formant. But sometimes the aorists defend to the contamination by the unmotivated round vowel,(i.e. -Ur): -Ir [present, aorist]:-Ar [future] vs. -Ir [present]:-Ur [aorist]:-Ar [future]; there are some another strategies for this by dialects;
seventhly, the very complicated question, but interesting and difficult to been described in few words; by the way, there is the fusion of "-Ip catır" form to the "-Ivatır" [in the colloquial speech; this is the first stage to possible more corruption, as in Oghuz branch; n.b. that "-Ip catır/etc.* as new praesens already has been grammaticalized (just because of its differentiating behaviour)] ; the verb catır[jatır] is interesting itself because it must be just cat[<√cat-(ıw)] as tur from √tur- but it is catır, possible explanations: there is an artefact of old aorist, or there is the contamination with otır, tur, cür forms.