Why did Russia accept the legal framework that led to Kosovo's independence?

I think a partial answer can be deduced from Russia's (2009) written statement to the ICJ proceedings on the Kosovo declaration of independence. In support of their view that 1244 implicitly prohibited a unilateral declaration of independence, Russia's statement recalled that Resolution 1244 reaffirmed the territorial integrity of Serbia:

"commitment of all Member States to the sovereignty and territorial integrity of the Federal Republic of Yugoslavia and the other States of the region, as set out in the Helsinki Final Act and annex 2"

The majority of ICJ didn't see this guarantee as having any impact on secession (by part of the country):

80. Several participants in the proceedings before the Court have contended that a prohibition of unilateral declarations of independence is implicit in the principle of territorial integrity. The Court recalls that the principle of territorial integrity is an important part of the international legal order and is enshrined in the Charter of the United Nations, in particular in Article 2, paragraph 4, which provides that: “All Members shall refrain in their international relations from the threat or use of force against the territorial integrity or political independence of any State, or in any other manner inconsistent with the Purposes of the United Nations.” In General Assembly resolution 2625 (XXV), entitled “Declaration on Principles of International Law concerning Friendly Relations and Co-operation among States in Accordance with the Charter of the United Nations”, which reflects customary international law (Military and Paramilitary Activities in and against Nicaragua (Nicaragua v. United States of America), Merits, Judgment, I.C.J. Reports 1986, pp. 101-103, paras. 191-193), the General Assembly reiterated “[t]he principle that States shall refrain in their international relations from the threat or use of force against the territorial integrity or political independence of any State”. This resolution then enumerated various obligations incumbent upon States to refrain from violating the territorial integrity of other sovereign States. In the same vein, the Final Act of the Helsinki Conference on Security and Co-operation in Europe of 1 August 1975 (the Helsinki Conference) stipulated that “[t]he participating States will respect the territorial integrity of each of the participating States” (Art. IV). Thus, the scope of the principle of territorial integrity is confined to the sphere of relations between States.

(My emphasis.)

Also Russia's position in their statement to the ICJ was that they understood/thought that the final status of Kosovo (cf. 1244) was not going to be decided unilaterally:

Paragraphs 11 (a) and (c) of Resolution 1244 mention that self-government and autonomy for Kosovo are to be ensured "pending a final/political settlement". [...] Yet a "settlement", both in its plain meaning and with specific reference to law and international relations, usually is something agreed upon by parties or decided by a competent authority. It is defined as "an agreement composing differences" or else as "an agreement ending a dispute or lawsuit". This understanding is particularly relevant in the context of the notion of "pacific settlement of disputes", where negotiation is considered as the first option to be pursued by the parties (Article 33 of the UN Charter). Moreover, in the case at hand, a clear reference to a negotiated settlement is contained in Resolution 1244 itself: "Negotiations between the parties for a settlement should not delay or disrupt the establishment of democratic self-governing institutions" (Annex 2, paragraph 8).

Apart from negotiation, Article 33 of the Charter lists, among the means of settlement of disputes, "enquiry, mediation, conciliation, arbitration, judicial settlement, resort to regional agencies or arrangements". All these means are characterized by a common feature: they envisage the involvement of a third party, duly authorized either to facilitate the negotiations or to decide on the matter. What this list excludes is a unilateral decision by one of the parties to the dispute. Therefore, even if one admits that Resolution 1244 does not exclude independence of Kosovo as a form of the "final settlement", such settlement was to be negotiated between the parties or, at the very least, to be decided upon by a body competent under international law to do so.

(Again my emphasis.)

And it goes on to detail (on several pages) why Russia thought the Security Council was the competent body for that determination, based e.g. on the trajectory of the Ahtisaari Plan. The ICJ rejected this view too, again based on the lack of specific clauses in 1244:

114. [...] In this regard the Court notes that contemporaneous practice of the Security Council shows that in situations where the Security Council has decided to establish restrictive conditions for the permanent status of a territory, those conditions are specified in the relevant resolution.. For example, although the factual circumstances differed from the situation in Kosovo, only 19 days after the adoption of resolution 1244 (1999), the Security Council, in its resolution 1251 of 29 June 1999, reaffirmed its position that a “Cyprus settlement must be based on a State of Cyprus with a single sovereignty and international personality and a single citizenship, with its independence and territorial integrity safeguarded” (para. 11). The Security Council thus set out the specific conditions relating to the permanent status of Cyprus. By contrast, under the terms of resolution 1244 (1999) the Security Council did not reserve for itself the final determination of the situation in Kosovo and remained silent on the conditions for the final status of Kosovo.

Still that doesn't really answer whether Russian diplomats, lawyers, and leaders really thought in 1999 they had obtain enough protection in Resolution 1244 against a unilateral declaration of independence of Kosovo. Maybe their Russian successors of 2009 just tried to salvage the situation (to their advantage) as best they could, given the cards they had been dealt.