A few incomplete thoughts on the question of mobilization. It won't solve many of the RU military's challenges in this war, but it could alter the dynamic. Fair to say that these are uncharted waters, and so we should take care with deterministic or definitive claims.
I wouldn't suggest that this can turn around Russia's fortunes in the war. However, I would take care being overly dismissive, especially looking out towards the medium term of this winter and 2023. Force availability and manpower matters, hence the implications can vary.
The Russian military has had structural manpower deficits throughout the war leading to problems with recruitment, retention, and rotation. Units can't be rotated, leading to exhaustion. Number of refuseniks grew. Hiring short term volunteers exacerbated retention issues.
Piecemeal solutions have led the Russian military to steadily cannibalize the force, using up officers, equipment, and enlisted professionals for reserve and volunteer units. Hence force quality degraded over time, as did morale, retention & exhaustion problems grew worse.
Mobilizing LDNR personnel, and using them to absorb losses led to a variegated force that lacked cohesion, interoperability, and suffered from weak morale. This approach seems to have largely exhausted itself in July, few men left to forcibly mobilize in LDNR.
The first and more important implication is not mobilization but enactment of stop-loss policies. Service contracts extended indefinitely, right to refuse deployment suspended, new criminal measures enacted to enforce what is a de facto introduction of wartime measures.
Caveat, this is an initial interpretation of the order. But it implies that you can no longer tear up your contract in the Russian military or leave service. Volunteers who signed up for short tours (4-6 months) are now extended for the duration of the mobilization period.
All mobilized personnel will be treated as contract servicemen, subject to these conditions. The situation with conscripts appears unchanged, but if Russia annexes these 4 UA regions, then it can technically deploy conscripts in those territories as well...
The optimal time for Russia to conduct mobilization was in April, before significant parts of the force and mobilization base were ineffectually consumed. So, what can this 'partial' mobilization achieve for Russia at this stage? The disappointing answer is it depends.
The first limit on mobilization is likely to be throughput - the system has to call-up, house, train, feed, equip, etc. Hence Shoigu's 300k number is likely to be notional, while actual mobilization proceed as a much more limited and phased process.
That said, I'm skeptical that mobilization infrastructure has sat entirely dormant. Russian voenkomats have been calling people up to update their info since April. Assembling reserve and volunteer battalions likely exercised some of this system already.
Since units typically train their personnel, its unclear what the capacity is in the system to absorb mobilized officers/soldiers, train them, and equip them. These are all uncertainties. Russian training of 3rd corps at Mulino might be an example of the approach (or not).
Hence mobilization is unlikely to generate new units for several months, and even then the output will be a lot less than what Moscow might expect. Mobilization is a coercive process in practice & economically disruptive. It also depends on how Russians choose to react.
However, RU mil could use mobilized personnel first to raise manning levels in currently deployed BTGs, many of which seem at 40-50%. Morale of mobilized personnel might be low, but individual replacements can start filling these units out faster than establishing new units.
Another approach might also be to deploy lower quality infantry regiments, akin to those currently seen among mobilized LDNR units, in order to hold large stretches of the line, i.e. the opposite of the 3rd corps effort to stand up a new volunteer formation with better kit.
The second main limitation stems from constraints on force employment. No matter how many personnel are mobilized, RU mil can only sustain and command a finite number of troops on the battlefield. Scaling has been one of the Russian military's chief problems in this war.
Russian capacity to implement partial mobilization is uncertain, as is the time it would take to produce results & how Russians will react to it. However, I'm also not sanguine on the proposition that it will make no difference. There's room for caution here.
Morale will continue to be an issue. Stop-loss policies may yield fewer refuseniks, but more deserters. Most UA advantages will remain. What partial mobilization may do in the coming months, depending on what actually comes of it, is help RU mil stabilize their lines.
This is in part why these coming months remain an important window of opportunity for UA to retake territory. Over the winter the contest will likely be one more defined by attrition and reconstitution. The extent to which mobilization can help RU reconstitute is unclear.
Mobilization comes with significant political risks and downsides for Moscow, but it could extend Russia's ability to sustain this war more so than alter the outcome. As always, these are just initial impressions and a very imperfect reading at best.
Perhaps a useful addition - mobilization & stop-loss might help Moscow stem the deteriorating quantity of the force, but not the deteriorating quality of the force & its morale. Having used up its best equipment, officers, & personnel, I don't see how this can be recovered.